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George  Washington  Flowers 
Memorial  Collection 

DUKE  UNIVERSITY  LIBRARY 


ESTABLISHED   BY  THE 

FAMILY  OF 

COLONEL  FLOWERS 


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INTELLECTUAL 


ARITHMETIC, 


VPON   THE 


INDUCTIVE  METHOD   OF  INSTRUCTION. 


BY  WARREN  COLBURN,  A.  M. 


WITH   AN  INTRODUCTION  TO   WRITTEN   ARITHMETIC,    BY   HIS   SON, 
WARREN   COLBURN. 


REVISED  AND  ADAPTED  TO  THE  USE  OF  SCHOOLS  IN  THE 
CONFEDERATE  STATES. 

BY    THOS.    O.    SUMMERS,    D.  D. 


-«•»- 


^^^r 


Nasfibtlle,  ^tnn. : 
SOUTHERN  METHODIST  PUBLISHING  HOUSE. 

18G2. 

u 


j^Tr^       INTRODUCTION,.  ^ 

The  firet  instructions  piven  to  the  child  in  Arithmetic,  are  usu- 
ally given  on  the  supposition  that  the  child  is  already  able  to  count. 
This  indeed  seems  a  sufficiently  low  requisition  ;  and  if  children 
were  taught  to  count  at  home  in  a  proper  manner,  they  would  have 
this  power  in  a  sufficient  degree  when  they  enter  the  primary 
scliool.  But  it  will  ))e  found  on  trial,  that  most  children,  wlien 
they  begin  to  go  to  school,  do  not  know  well  how  to  count.  This 
may  be  proved  by  requiring  them  to  count  20  beans  or  grains  of 
corn.  Few  of  them  will  do  it  without  mistake.  The  difficulty  is, 
they  have  been  taught  to  repeat  the  numerical  names,  one,  two, 
three,  in  order,  without  attaching  ideas  to  them.  They  learn  to 
count  without  counting  things.  This  point,  then,  calls  for  the 
teacher's  first  attention— to  lead  the  child  to  apprehend  the.  mean- 
ing of  each  numerical  word  by  using  it  in  connection  with  objects. 

The  kind  of  objects  to  be  employed  as  counters  should,  of  course, 
be  similar,  as  marks  on  the  blackboard,  beans,  pieces  of  wood,  or 
o^cork,  or  the  balls  in  a  numeration  frame.  Provided  they  are 
similar,  and  large  enough  to  be  seen  without  effort  by  all  the  class, 
it  is  of  little  consequence  wliat  they  are  :  the  simpler  the  better, 
and  those  which  the  teacher  devises  or  makes,  will,  other  things 
being  equal,  be  best  of  all.  Not  more  than  ten  should  be  used  or 
exhibited  to  the  children  in  the  first  few  lessons. 

LESSON  I. 

Let  the  class  have  their  attention  called  to  the  teacher ;  and 
when  he  lays  down  a  counter,  where  all  can  see  it,  let  them  say 
one;  let  the  teacher  lay  down  another,  and  the  class  say  two;  and 
so  on  up  to  ten.  If  any  of  the  class  become  inattentive,  let  the 
teacher  stop- at  once:  and,  after  the  attention  is  fully  centred  on 
him,  let  him  begin  again. 

After  going  through  this  addition  a  few  times  in  this  form,  it 
may  be  varied  thus  :  The  teacher  laying  down  the  counters,  one 
by  one,  as  before,  the  class  may  be  led  to  say,  one  and  one  are 
two,  two  and  one  are  three,  three  and  one  are  four,  etc. 

The  above  mode  of  adding  may  be  shortened  by  leading  the 
class  to  say  as  follows :  One  and  one  are  two,  and  one  are  three, 
and  one  are  four,  etc. 

At  any  time  the  word  designating  the  counter  may  be  used  along 
with  the  number,  as  beans,  balls,  pieces,  marks,  or  books,  as  the 
case  may  be. 

At  times,  it  will  be  well  to  give  some  fictitious  designation  to 
the  counters,  such  as  the  teacher,  or  still  better,  such  as  some  one 
of  the  class  may  choose,  calling  them  men,  sheep,  horses,  etc. 


f^^^V 


IV  INTRODUCTORY   LESSONS. 

Next  to  Addition,  as  illustrated  above,  should  come  Subtraction. 
Having  counted  ten,  let  the  teacher  take  away  one,  and  the  class 
be  made  to  say,  one  from  ten  leaves  nine,  one  from  nine  leaves 
eight,  etc.  In  Subtraction,  the  same  variations  may  be  introduced 
as  in  Addition.  No  further  illustrations  of  this  operation  need  be 
given,  as  the  teacher's  discretion  will  supply  all  that  is  necessary. 

In  connection  with  these  exercises,  let  the  pupil  be  taught  to  re- 
peat in  reversed  order  the  numerical  words  they  have  employed, 
counting  from  one  up  to  ten,  and  then  in  reverse  order  from  ten  to 
one. 

It  is  not  to  be  supposed  that  the  whole  of  the  foregoing  lesson 
can  be  learned  at  one  exercise.  It  is  only  a  small  part  of  it  that 
children  will  at  first  have  sufficient  power  of  attention  to  go  over 
with  profit.  The  same  remark  may  be  made  respecting  the  fol- 
lowing Introductory  Lessons. 

LESSON  II. 

Let  the  teacher  call  the  attention  of  the  class,  and  require  them 
to  count,  and  then  lay  down,  one  by  one,  a  small  number  of  count- 
ers, say,  for  example,  five ;  then  let  him  separate  them  into  two 
parts,  as  one  and  iour,  thus,  •  •  •  •  •,  and  say,  "  one  and  four 
are  five,  "  and  require  the  class  to  say  the  same.  Then  let  him 
divide  the  number  into  different  parts,  as  two  and  three,  three  and 
two,  four  and  one,  one  and  one  and  three,  etc.,  requiring  the  clasa 
with  each  division  to  name  the  parts  and  make  the  addition.  Let 
them  always  begin  at  the  left  end  of  the  line  of  counters  as  they 
face  them.  Having  exhausted  the  combinations  of  five,  let  the 
number  six  be  taken,  giving  combinations  like  the  following  :  •  • 

•  ••      •      ••-;  etc. 

It  may  be  found  that  a  lower  number  than  ftve  should  be  made 
the  first  step  in  this  exercise. 

After  the  combinations  of  six  have  been  exhausted,  the  number 
seven  may  bo  taken,  and  then  successively,  eight,  nine,  and  ten. 

As  a  part  of  this  Lesson,  each  question  in  addition  should  be 
converted  into  a  question  in  subtra(;tion ;  thus  five  and  three  are 
eight;  then,  liaving  put  the  two  parts  together,  which  make  eight, 
remove  the  three,  and  lead  the  class  to  say,  '■'three  from  eight 
leaves  five,'' 

The  following  exercise  is  important  in  this  connection.  Let  the 
teacher  select  s^me  number,  and  give  one  part  of  it,  and  require 
the  class  as  quick  as  possible  to  name  the  complementary  part. 
Thus  let  six  be  the  number,  the  exercise  will  be  as  follows : 
Teacher:  "Now  attend,  six  is  the  numl)er;  I  am  going  to  name 
one  part  of  it ;  when  you  hear  me  name  it,  do  you  all  name  the 
other  part  as  quick  as  you  can;  now  be  ready;  five."  Class: 
''One."  Teacher:  "FoMr."  Class:  "Tmjo."  Teacher:  "TArec." 
Class:  "T/irfe."     Teacher :  "One  "     Class :  "F?'w,"  etc. 

This  exercise  should  not  be  pressed  too  fast,  but  carried  on  gra- 
dually, as  the  pupil's  strength  of  mind  will  allow.  Special  pains 
should  be  taken  that  the  number  ten  be  perfectly  mastered  in  this 
form  of  combining  its  parts.  This  will  give  the  pupil  the  most  im- 
portant aid  in  all  his  calculations  in  larger  numbers. 


INTRODUCTOEY    LESSONS. 


LESSON   III. 


For  a  number  of  days  after  becrinning  the  foregoing  exercises, 
the  child  should  not  have  the  book  at  all  in  his  hands.  If  the  child 
has  the  book  in  his  possession,  it  will  be  well  for  the  teacher  to 
take  it  for  a  few  days,  and  let  the  pui)il  employ  himself  at  his  seat 
in  writing  on  a  slate,  or  with  ot])er  books.  In  this  way,  the  child 
has  awakened  within  him  the  idea  of  calculation  in  numbers,  with- 
out having  become  wearied  with  the  reading  of  what  excites  no 
interest.  After  a  few  days,  however,  the  book  may  be  put  into  tlie 
pupil's  hands,  aud  he  may  be  directed  to  get  a  lesson  in  Section  I. 
In  the  meanwhile,  the  Introductory  Lessons  should  be  continued, 
and  form  a  part  of  each  day's  exercise  tiil  the}'  are  finished.  In  this 
way,  the  pupil,  in  studying  his  first  lesson  from  the  book,  will  al- 
ready have  learned  the  use  of  counters,  and  will  naturally  resort  to 
them  at  his  seat,  using  beans  or  marks  on  his  slate  for  this  purpose. 
It  will  be  far  better  for  him  to  come  to  the  use  of  counters  in  this 
natural  way.  than  to  be  enjoined  to  use  them  before  he  has  been 
interested  in  witnessing  their  application. 

The  pupil,  in  the  preceding  lef«sons,  has  become  acquainted  with 
all  the  nujnbers  as  far  as  ten,  regarding  them  either  as  units,  or  as 
grou])ed  into  i)art8  of  a  larger  whole.  The  next  step  is  to  carry 
him  through  the  numbers  from  ten  to  twenty. 

First,  let  the  class  count  with  the  objects  before  them  from  one 
up  to  twenty  ;  then,  removing  all  but  ten,  let  the  ten  be  grouped 
in  a  pile  ;  or  if  they  are  marks  on  the  board,  let  them  be  enclosed 
by  a  line  drawn  around  them,  and  begin  to  count  upward  from  ten, 
"  One  and  ten  are  eleven ;  two  and  ten  are  twelve  ;  three  and  ten 
are  thirteen."  Here  pause,  and  examine  the  composition  of  the 
word,  thirteen — three  ten,  or  three  and  ten.  Show  how  the  three 
is  spelt  in  thirteen,  and  also  how  the  ten  is  spelt.  Then  proceed 
"four  aud  ten  are  fourteen,"  examining  the  word  as  in  the  former 
case;  '"five  and  ten  are  fifteen;  six  and  ten  are" — perhaps  some 
one  in  the  class  will  now  be  able  to  give  the  compound  word  ;  then 
go  on,  "seven  and  ten,  eight  and  ten,  nine  and  ten,  ten  and  ten." 

When  they  can  give  the  compound  words  readily  from  the  sim- 
ple ones,  then  give  them  the  compound  word,  and  let  the  class 
separate  it  into  its  two  component  words ;  thus :  Teacher : 
"Seventeen."  Class:  "Seven  and  ten,"  etc.  Thus  far  let  the 
teacher  be  careful  to  present  the  name  of  the  smaller  of  the  two 
numbers  first,  for  that  is  the  order  in  which  the  compound  word 
presents  them ;  let  the  teacher  say  four  and  ten,  and  not  ten  and 
four.  After  the  class  have  caught  the  analogy  between  the  simple 
words,  and  the  compounds  which  they  form,  so  that  one  instantly 
suggests  to  them  the  other,  then  the  order  of  the  words  may  be 
changed,  and  the  ten  put  first.  The  caution  here  suggested  may 
seem  to  some  unnecessary ;  but  a  careful  observation  of  the  men- 
tal habits  of  children  will  not  fail,  I  think,  to  show  its  importance. 

In  the  analysis  of  the  compound  words  from  ten  to  twenty, 
eleven  and  twelve  should  be  omitted  till  the  last;  for  as  the  simple 
words  of  wliich  the}'  are  foiimed  are  disguised  or  obsolete,  they 
tend  to  obscure,  rather  than  elucidate  the  subject  to  the  mind  of  a 
child.  Having  obtained  the  idea  through  the  other  words  in  the 
series,  he  may  take  the  statement  respecting  these  on  trust. 


VI  INTRODUCTORY   LESSONS. 


LESSON  IV. 


Having  counted  twenty,  and  grouped  the  number  in  two  tens, 
let  the  class  count  ten  more,  making  in  all  thirty,  or  three  tens. 
Keeping  the  tens  separate,  let  the  class  count  ten  more,  making 
forty,  or  four  tens.  Let  the  class  then  answer  such  questions  as 
the  following:  Twenty  are  how  many  tens?  Thirty  are  how  many 
tens?  Forty  are  how  many  tens?  Four  tens  are  what  number? 
Three  tens  are  what  number?    Two  tens  are  what  number? 

After  this,  they  may  proceed  with  the  higher  multiples  of  ten, 
fifty,  sixty,  seventy,  eighty,  ninety,  a  hundred. 

Through  the  whole  of  this  exercise,  each  multiple  of  ten  should 
be  presented  in  groups  of  ten,  so  as  to  aid  the  idea  by  the  visible 
representation. 

The  pupils  should  be  led  to  see  the  significancy  of  each  numer- 
ical name ;  that  thirty- seven,  for  example,  means  three  tens  and 
seven;  fifty-six  means  live  tens  and  six. 

In  this  way  the  pupil  may  be  led  to  understand  the  Decimal 
Ratio,  at  this  early  stage,  and  no  farther  trouble  need  be  taken  in 
that  direction.  When  in  a  latter  stage  of  study,  he  comes  to  the 
Decimal  notation  in  written  Arithmetic,  he  will  find  it  only  a  natural 
mode  of  expressing  ideas  already  rendered  familiar  in  practice. 

LESSON  V. 

Let  the  teacher  stand  at  the  board,  and  call  the  attention  of  the 
class  to  what  he  shall  write  ;  then,  making  two         f     | 
marks,  ask,  "How  many  marks  on  the  board?"  | 

When  the  class  have  answered,  let  the  teacher  j 

write   two  more,  and  ask,  "How  many  now?"  I 

and   so  on   to  the   number  of  twelve  or  more.         |      \ 
Then  take  a  writing-book  or  sheet  of  paper,  and  | 

covering  all  but  two  of  the  marks,  let  the  class  j 

repeat  the  same  process  while  the  teacher  re-  [  I 
moves  the  book,  so  as  to  bring  two  more  into  view  at  each  re- 
move ;  the  numbers  read  by  the  class  being  two,  four,  six,  eight, 
ten,  etc.  Then  let  the  process  be  reversed,  subtracting  two 
successively,  which  gives,  beginning  with  sixteen,  the  following : 
sixteen,  fourteen,  twelve,  ten,  etc. 

Again,  the  teacher  may  say  to  the  class,  "When  I  made  those 
marks,  how  many  did  I  make  at  a  time?"  Class:  "Two." 
Teacher  :  "Did  I  make  two  more  than  once?"  Class:  "Yes,  sir, 
a  good  many  times."  Then  the  teacher,  covering  up  all  but  two  : 
"Now  look,  how  many  times  two  are  there?"  Class:  "Once." 
Teacher:  "Once  two  are  how  many?"  Then,  after  the  class 
have  answered,  showing  two  more,  "How  many  times  two  do  you 
see?"  "Twice  two  are  how  many?"  Then  go  on  in  the  same 
way  with  three  twos,  four  twos,  etc.,  to  the  end. 

At  this  point  the  pupils  may  be  taught  the  distinction  between 
even  and  odd  numbers,  and  be  trained  to  repeat  rapidly  the  even 
numbers,  from  two  up  to  twenty. 

The  pupils  may  derive  important  aid  in  adding  and  multiplying, 
by  grouping  the  numerical  names  with  the  voice,  in  something 
like  the  following  manner:  Teacher:  "Listen  now  to  me;  one^ 


ISTEODUCTOBY    LBSSONS.  Vll 

two— three,  four— five,  six.  How  many  twos  did  I  count?"  Class : 
"Three  twos."  Teacher:  '•  Count  three  twos  just  as  I  did." 
Then  let  the  teacher  ask,  "Three  times  two  are  how  many?" 
Then  require  them  to  count  four  twos,  five  twos,  etc.  This  method 
may  be  brought  in  aid  of  all  their  earlier  attempts  in  adding  and 
muitiplving. 

Care  'should  be  taken  in  this  exercise  tliat  the  words  be  pro- 
nounced in  a  quick  and  neat  manner,  with  a  distinct  pause  between 
the  groups.  If  rightly  conducted,  this  exercise  furnishes  one  of 
the  best  aids  for  overcoming  the  habit  of  counting,  instead  of  ad- 
ding—that fatal  clog  whicli  keeps  back  many  scholars  in  arith- 
metic from  making  any  rapid  progress. 

rr,,  ■,,..■  r  J.        1      •     •  r     I  I  eleven. 

The  successive  additions  of  two,  beginning  .  j  ^^^^ 

with  one,  furnish  the  series  of  odd  numbers.  !   j  ggyg'u^         « 

This  and  the  preceding  exercise  exhibit  all  the     ■!      .  |  five 
results  made  by  the  additions  of  the  number  j  .  ^]^yQQ 

t^o-  [         I  one.  ' 

The  pupil  should,  in  connection  wMth  this  exercise,  be  taught  to 
group  the  numerical  names,  by  twos,  beginning  from  one,  so  as  to 
furnish  the  series  of  odd  numbers,  thus :  one— two,  three— fooir, 
five,  etc. 

Finally,  let  him  count  the  even  series,  up  to  twenty,  and  then 
down,  till  he  can  do  it  readily. 

Let  him  count  the  odd  series  up  to  twenty-one,  up  and  down. 

LESSON  VL 
To  teach  the  pupils  to  add  the  number  three,  let  the  teacher  make 


three  marks  on  the  board,  and  require  the 
class   to   name    the    number;    then   three 


I  I  I  fifteen. 
I  I  I  twelve. 


more,  and  so  on.     Then,  covering  all  but        <        111  nine 
the   lower  three,  let  the   class   name  the  I  I  I  six. 

number  in  view,  as  the   teacher  exhibits        i       III  three, 
successively  three  more  at  each  remove,  or  three  less. 

The  pui)ils  may  then  be  led  to  count  in  groups  of  three;  thus: 
one,  two,  three— four,  five,  six — sevei\,  eight,  nine,  etc. 

From  this,  they  may  be  readily  led  to  name  the  multiples  of 
three ;  three,  six,  nine,  twelve,  etc. 

The  first  variation  from  the  above  method  in  the  addition  of  three 
may  be  made  by  beginning  with  two,  which        f     ||  i  fourteen, 
gives  the  following  numbers.     In  order  to  j  |  |  eleven, 

secure  the  necessary  amount  of  repetition        ■      i  i  i  eight. 
in  these  exercises,  and  yet  to  avoid  weari-  i  i  i  five, 

ness,  it  will  be  well  often  to  change  the  [  t  I  two. 
manner  of  adding.  Thus,  in  the  accompanying  example,  the  pupils 
may  simply  name  the  series  made  by  the  successive  additions  ;  as 
two,  five,  eight,  eleven,  fourteen ;  or,  they  may  state  the  process 
more  fully,  thus:  two  and  three  are  five,  and  three  are  eight,  and 
thi'ee  are  eleven,  etc.  A  still  fuller  statement  of  the  operation 
would  be  thus:  two  and  three  are  five,  five  and  three  are  eight, 
eight  and  three  are  eleven,  etc.    Tiie  only  remaining  variety  in 


viii 


INTBODUOTORT   LESSONS. 


adding  three  will  be  exhibited  by  beginning  with  one,  which  may 
be  illustrated  in  the  same  way  as  the  others,  giving  the  numbers, 
one,  four,  seven,  ten,  etc. 

LESSON  vn. 

After  the  numbers  have  been  mastered  in  the  manner  above  sug- 
gested, thev  should  be  combined  in  a  variety  of  ways  by  means  of 
marks  on  the  board.  The  columns  of  marks  subjoined  are  giveu 
merely  as  examples  and  hints  on  this  point. 

I  I 
I 

1  I 
I  I 


1  I 
I 


A  single  column  may  be  made  to  furnish  exercise  suflScient  for  a 
recitation ;  for,  by  adding  or  erasing  a  single  mark  at  the  begin- 
ning, a  change  is  made  through  the  whole. 


1     1  1 
1         III 
1        III 

1         1  1 

111             I 
II       .11 
i  1            i 

III      III 

1  1  1 

1        1  i 

1         1 1 

1         1 

III       III 

i     Mi 
ill       1 

I  1  1 

1        III 
1        III 

1         1 

III       1 
1      1  1 

II      11 
11     III 

1  1  1 

1          1 

1          1 1 

1  1     III 
III       1 

ARITHMETIC. 


PART    I. 

SECTION    I 


A.*  1.  How  many  thumbs  have  you  on  your  right 
hand?  how  many  on  your  left?  how  many  on  both 
together  ? 

2.  How  many  hands  have  you? 

3.  If  you  have  two  nuts  in  one  hand,  and  one  in  the 
other,  how  many  have  you  in  both  * 

4.  How  many  fingers  have  you  on  one  hand  ? 

5.  If  you  count  the  thumb  with  the  fingers,  how 
many  will  it  make  ? 

6.  If  you  shut  your*  thumb  and  one  finger,  and  leave 
the  rest  open,  how  many  will  be  open  ? 

-   7.  If  you  have  two  cents  in  one  hand,  and  two  in 
the  other,  how  many  have  you  in  both  ? 

8.  James  has  two  apples,  and  William  has  three ; 
if  James  gives  his  apples  to  William,  how  many  will 
William  have  ? 

9.  If  you  count  all  the  fingers  on  one  hand,  and 
two  on  the  other,  how  many  will  there  be  ? 

10.  George  has  three  dimes,  and  Joseph  has  four; 
how  many  have  they  both  together  ? 

11.  Robert  gave  five  cents  for  an  orange,  and  two 
for  an  apple ;  how  many  did  he  give  for  both  ? 

12.  If  a  custard  costs  six  cents,  and  an  apple  two 
cents,  how  many  cents  will  it  take  to  buy  an  apple  and 
a  custard  ? 

13.  If  you  buy  a  pint  of  nuts  for  five  cents,  and  an 
orange  for  three  cents,  how  many  cents  would  you  give 

*  The  first  questions  in  this  section  are  intended  for  very  young 
children.  It  will  be  well  for  the  instruotor  to  give  a  great  many 
more  of  this  kind.    Older  pupils  may  prqjt  tjiese. 

(9) 


10  ARITHMETIC.  [part  i. 

for  both  ?  how  many  more  for  the  nuts  than  for  the 
orange  ? 

14.  If  an  ounce  of  figs  is  worth  six  cents,  and  a  half 
a  pint  of  cherries  is  worth  three  cents,  how  much  are 
they  both  worth  ? 

15.  Joseph  had  five  plums,  and  John  gave  him  four 
more ;  how  many  had  he  then  ? 

16.  How  many  fingers  have  you  on  both  hands  ? 

17.  How  many  fingers  and  thumbs  have  you  on  both 
hands  ? 

18.  If  you  had  six  marbles  in  one  hand,  and  four  in 
the  other,  how  many  would  you  have  in  the  one 
more  than  in  the  other  ?  how  many  would  you  have 
in  both  hands  ?       * 

19.  David  had  seven  nuts,  and  gave  three  of  them 
to  George  ;  how  many  had  he  left? 

20.  Two  boys,  James  and  Robert,  played  at  marbles  ', 
when  they  began,  they  had  seven  apiece,  and  when 
they  had  done,  James  had  won  four;  how  many  had 
each  then  ? 

21.  A  boy,  having  eleven  nuts,  gave  away  three  of 
them  ;  how  many  had  he  left  ? 

22.  If  you  had  eight  dimes  and  your  papa  should 
give  you  five  more,  how  many  would  you  have  ? 

23.  A  man  bought  a  sheep  for  eight  dollars,  and  a 
calf  for  seven  dollars ;  what  did  he  give  for  both  ? 

24.  A  man  bought  a  barrel  of  flour  for  eight  dollars, 
and  sold  it  for  four  dollars  more  than  he  gave  for  it; 
how  much  did  he  sell  it  for. 

25.  A  man  bought  a  hundred  weight  of  sugar  for 
nine  dollars,  and  a  barrel  of  flour  for  seven  dollars; 
how  much  did  he  give  for  the  whole  ? 

26.  A  man  bought  three  barrels  of  cider  for  eight 
dollars,  and  ten  bushels  of  apples  for  nine  dollars; 
how  much  did  he  give  for  the  whole  ? 

27.  A  man  bought  a  firkin  of  butter  for  twelve 
dollars,  but,  it  being  damaged,  he  sold  it  again  for 
eight  dollars ;  how  much  did  he  lose  ? 


SEO.  i.J  ARITHMETIC.  H 

28.  A  man  bought  three  sheep  for  fifteen  dollars, 
but  could,  not;  sell  them  ^gain  for  so  much  by  ei"-ht 
dollars ;  how  much  did  he  sell  them  for  ? 

29.  A  man  bought  sixteen  pounds  of  coffee,  and 
lost  seven  pounds  of  it  as  he  was  carrying  it  home  • 
how  much  had  he  left?  ' 

30.  A  man  bought  nineteen  pounds  of  sugar,  and 
having  lost  a  part  of  it,  he  found  he  had  nine  pounds 
left;  how  much  had  he  lost? 

31.  A  man,  owing  fifteen  dollars,  paid  nine  dollars 
of  it ;  how  much  did  he  then  owe  ? 

32.  A  man,  owing  seventeen  dollars,  paid  all  but 
seven  dollars ;  how  much  did  he  pay  ? 

B.  1.  Two  and  one  are  how  many? 

2.  Two  and  two  are  how  many  ? 

3.  Three  and  two  are  how  many  ? 
-4.  Four  and  two  are  how  many  ? 

6.  Five  and  two  are  how  many  ? 

6.  Six  and  two  are  how  many? 

7.  Seven  and  two  are  how  many  ? 

8.  Eight  and  two  are  how  many  ? 

9.  Nine  and  two  are  how  many  ? 

10.  Ten  and  two  are  how  many  ? 

11.  Two  and  three  are  how  many? 

12.  Three  and  three  are  how  many? 

13.  Four  and  three  are  how  many  ? 

14.  Five  and  three  are  how  many  ? 

15.  Six  and  three  are  how  many  ? 

16.  Seven  and  three  are  how  many? 

17.  Eight  and  three  are  how  many? 

18.  Nine  and  three  are  how  many  ? 

19.  Ten  and  three  are  how  many  ? 

20.  Two  and  four  are  how  many  ? 

21.  Three  and  four  are  how  many  ? 

22.  Four  and  four  are  how  many  ? 

23.  Five  and  four  are  how  many  ? 

24.  Six  and  four  arc  how  many  ? 


12  ARITHMETIC.  [parti. 

25.  Seven  and  four  arc  how  many  ? 

26.  Eight  and  lour  are  how  many?  ■ 

27.  Nine  and  four  are  how  many  ? 

28.  Ten  and  four  arc  how  many? 

29.  Two  and  five  are  how  many  ? 

30.  Three  and  five  arc  how  many  ? 

31.  Four  and  five  are  how  many? 

32.  Five  and  five  are  how  many  ? 
38.  Six  and  five  are  how  many? 

34.  Seven  and  five  are  how  many  ? 

35.  Eight  and  five  are  how  many  ? 

36.  Nine  and  five  arc  how  many  ? 

37.  Ten  and  five  are  hoAV  many  ? 

38.  Two  and  six  are  how  many? 

39.  Three  and  six  are  how  many? 

40.  Four  and  six  are  how  many  ? 

41.  Five  and  six  are  how  many? 

42.  Six  and  six  are  how  many  ? 

43.  Seven  and  six  are  how  many? 

44.  Eight  and  sik  are  how  many? 

45.  Nine  and  six  are  how  many  ? 

46.  Ten  and  six  are  how  many  ? 

47.  Two  and  seven  are  how  many  ? 

48.  Three  and  seven  are  how  many  ? 

49.  Four  and  seven  are  how  many? 

50.  Five  and  seven  are  how  many  ? 

51.  Six  and  seven  are  how  many  ? 

52.  Seven  and  seven  are  how  many  ? 

53.  Eight  and  seven  are  how  many  ? 

54.  Nine  and  seven  are  how  many  ? 

55.  Ten  and  seven  are  how  many  ? 

56.  Two  and  eight  are  how  many  ? 

57.  Three  and  eight  are  how  many  ? 

58.  Four  and  eight  are  how  many? 

59.  Five  and  eight  are  how  many? 

60.  Six  and  eight  arc  how  many  ? 

61.  Seven  and  eight  are  how  many? 

62.  Eight  and  eight  are  how  many  ? 


SEC.  i]  ARITHMETIC.  13 

63.  Nine  and  eight  are  how  many? 

64.  Ten  and  eight  are  how  many  ? 

65.  Two  and  nine  are  how  many  ? 

66.  Three  and  nine  are  how  many  ? 

67.  Four  and  nine  are  how  many? 

68.  Five  and  nine  are  how  many? 

69.  Six  and  nine  are  how  many  ? 

70.  Seven  and  nine  are  how  many? 

71.  Eight  and  nine  are  how  many? 

72.  Nine  and  nine  are  how  many? 

73.  Ten  and  nine  are  how  many  ? 

74.  Two  and  ten  are  how  many  ? 

75.  Three  and  ten  are  how  many  ? 

76.  Four  and  ten  are  how  many  ? 

77.  Five  and  ten  are  how  many  ? 

78.  Six  and  ten  are  how  many  ? 
,79.   Seven  and  ten  are  how  many? 

80.  Eight  and  ten  are  how  many  ? 

81.  Nine  and  ten  are  how  many  ? 

82.  Ten  and  ten  are  how  many  ? 

C.  1.  Two  and  one  are  how  many? 

2.  Two  and  two  are  how  many  ? 

3.  Three  and  two  are  how  many  ? 

4.  Five  and  two  are  how  many  ? 

5.  Four  and  two  are  how  many  ? 

6.  Six  and  two  are  how  many  ? 

7.  Eight  and  two  are  how  many  ? 

8.  Five  and  three  are  how  many? 

9.  Seven  and  three  are  how  many? 

10.  Four  and  three  are  how  many? 

11.  Two  and  three  are  how  many? 

12.  Two  and  six  are  how  many  ? 

13.  Two  and  eight  are  how  many? 

14.  Six  and  three  are  how  many  ? 

15.  Three  and  four  are  how  many  ? 

16.  Three  and  six  are  how  many? 

17.  Two  and  seven  are  how  many? 


ARITHMETIC.  [paut  i. 

18.  Ten  and  two  are  how  many  ? 

19.  Two  and  four  are  how  many  ? 

20.  Three  and  seven  are  how  many  ? 

21.  Four  and  four  are  how  many  ? 

22.  Five  and  four  are  how  many  ? 

23.  Seven  and  two  are  how  many  ? 

24.  Two  and  five  are  how  many  ? 

25.  Three  and  three  are  how  many  ? 

26.  Four  and  five  are  how  many  ? 

27.  Nine  and  two  are  how  many? 

28.  Three  and  five  are  how  many? 

29.  Two  and  ten  are  how  many  ? 

30.  Three  and  eight  are  how  many  ? 

31.  Ten  and  three  are  how  many? 

32.  Two  and  nine  are  how  many  ? 
83.  Four  and  six  are  how  many  ? 

34.  Eight  and  three  are  how  many  ?  ^ 

35.  Seven  and  four  are  how  many  ? 

36.  Nine  and  three  are  how  many  ? 

37.  Six  and  four  are  how  many  ? 

38.  Five  and  five  are  how  many  ? 

39.  Three  and  nine  are  how  many  ? 

40.  Four  and  seven  are  how  many  ? 

41.  Six  and  five  are  how  many  ? 

42.  Three  and  ten  are  how  many  ? 

43.  Eight  and  four  are  how  many  ? 

44.  Five  and  eight  are  how  many  ? 

45.  Four  and  nine  are  how  many  ? 

46.  Five  and  six  are  how  many  ? 

47.  Ten  and  four  are  how  many? 

48.  Seven  and  five  are  how  many  ? 

49.  Six  and  six  are  how  many  ? 

50.  Nine  and  four  are  how  many  ? 

51.  Eight  and  five  are  how  many  ? 

52.  Five  and  nine  are  how  many  ? 

53.  Four  and  ten  are  how  many  ? 

54.  Six  and  seven  are  how  many  ? 

55.  Four  and  eight  are  how  many  ? 


BKO.  1.]  ARITHMETIC.  ^-  16 

56.  Nine  and  five  are  how  many  ? 

57.  Six  and  eight  are  how  many  ? 

58.  Ten  and  five  are  how  many  ? 

59.  Seven  and  six  are  how  many  ? 

60.  Eight  and  seven  are  how  many  ? 

61.  Six  and  nine  are  how  many  ? 

62.  Seven  and  seven  are  how  many  ? 

63.  Eight  and  six  are  how  many  ? 

64.  Ten  and  six  are  how  many  ? 

65.  Eight  and  eight  are  how  many  ? 

66.  Nine  and  seven  are  how  many? 

67.  Ten  and  eight  are  how  many  ? 

68.  Six  and  ten  are  how  many  ? 

69.  Five  and  seven  are  how  many  ? 

70.  Nine  and  six  are  how  many  ? 

71.  Seven  and  eight  are  how  many  ?         • 

72.  Eight  and  nine  are  how  many  ? 

73.  Nine  and  nine  are  how  many  ? 
71.  Five  and  ten  are  how  many  ? 

75.  Seven  and  nine  are  how  many  ? 

76.  Nine  and  eight  are  how  many  ? 

77.  Eight  and  ten  arc  how  many  ? 

78.  Ten  and  nine  are  how  many  ? 

79.  Seven  and  ten  are  how  many  ? 

80.  Nine  and  ten  are  how  many? 

81.  Ten  and  ten  are  how  many  ? 

D.  1.  Three  boys,  Peter,  John,  and  Oliver,  gave 
some  money  to  a  beggar.  Peter  gave  seven  cents,  John, 
four  cents,  and  Oliver  three  cents ;  how  many  did  they 
all  give  him  ? 

2.  How  many  did  Peter  give  more  than  Oliver  ? 

3.  Frank  had  nine  pears,  and  gave  three  of  them  to 
Harry  ;  how  many  had  he  left ;  and  how  many  more 
than  Harry  had  he  then  ? 

4.  John  had  ten  peaches,  Harry  twelve,  and  Charles 
thirteen;  John  gave  three  to  Stephen,  Harry  gave 
him  six,  and  Charles  gave  him  five ;  how  many  had 
Stephen  ?  and  how  many  had  each  left  ? 


16  ARITHMETIC.  [part  i. 

5.  A  boy  had  twenty  apples,  and  gave  them  to  his 
companions,  as  follows  :  to  one  he  gave  three ;  to  an- 
other, two;  to  another  four;  and  to  another,  five;  how 
many  did  he  give  away  ?  and  how  many  had  he  left  ? 

6.  A  boy  gave  to  one  of  his  companions  eight 
peaches;  to  another,  six;  to  another,  four;  and  kept 
two  himself;  how  many  had  he  at  first? 

7.  A  boy  went  to  the  confectioner's  and  bought 
three  cakes  of  gingerbread,  for  which  he  gave  a  cent 
apiece ;  two  buns,  for  which  he  gave  three  cents 
apiece;  one  custard  for  four  cents,  and  one  orange 
for  six  cents ;  how  many  cents  did  he  spend  for  the 
whole  ? 

8.  A  boy  having  twenty-five  cents,  bought  one 
quart  of  cherries  for  eight  cents,  one  orange  for  six 
cents,  and  gave  away  three  cents;  how  many  cents 
had  he  left  ? 

9.  A  boy  bought  a  box  for  eighteen  cents,  and  gave 
eight  cents  to  have  it  painted,  and  then  sold  it  for 
thirty-two  cents;  how  much  did  he  gain  by  the 
bargain  ? 

10.  A  man  bought  a  sleigh  for  seventeen  dollars, 
and  gave  nine  dollars  to  have  it  repaired  and  painted, 
and  then  vsold  it  for  twenty-three  dollars ;  how  much 
did  he  lose  by  the  bargain  r* 

11.  Eleven  and  two  are  how  many? 

12.  Eleven  and  three  arc  how  many  ? 

13.  Eleven  and  four  are  how  many? 

14.  Eleven  and  five  are  how  many? 

15.  Eleven  and  six  are  how  many? 

16.  Eleven  and  seven  are  how  many? 

17.  Eleven  and  eight  are  how  inany? 

18.  Eleven  and  nine  arc  how  many? 

19.  Eleven  and  ten  are  how  many? 

20.  Twelve  and  two  are  how  many? 

21.  Twelve  and  three  are  how  many? 

22.  Twelve  and  four  are  how  many? 


fiBC.  I.]  AttlTHMETIC.  17 

23.  Twelve  and  five  are  how  many? 

24.  Twelve  and  six  arc  how  many? 

25.  Twelve  and  seven  are  how  many? 

26.  Twelve  and  eight  are  how  many? 

27.  Twelve  and  nine  are  how  many? 

28.  Twelve  and  ten  are  how  many? 

29.  Thirteen  and  two  are  how  many? 

30.  Thirteen  and  three  are  how  many? 

31.  Thirteen  and  four  are  how  many? 

32.  Thirteen  and  five  are  how  many? 

33.  Thirteen  and  six  are  how  many? 

34.  Thirteen  and  seven  are  how  many  2 

35.  Fourteen  and  two  are  how  many? 

36.  Fourteen  and  three  are  how  many? 

37.  Fourteen  and  four  are  how  many? 

38.  Fourteen  and  five  are  how  many? 

39.  Fourteen  and  six  are  how  many  ? 

40.  Fifteen  and  two  are  how  man}'? 

41.  Fifteen  and  three  are  how  many? 

42.  Fifteen  and  four  are  how  many? 

43.  Fifteen  and  five  are  how  many? 

44.  Sixteen  and  two  are  how  many? 

45.  Sixteen  and  three  are  how  many? 

46.  Sixteen  and  four  are  how  many? 

47.  Seventeen  and  two  are  how  many? 

48.  Seventeen  and  three  are  how  many? 

49.  Eighteen  and  two  are  how  many? 

E.  1.  A  man  hought  a  sheep  for  nine  dollars,  and 
to  pay  for  it  he  gave  five  bushels  of  corn  worth  four 
dollars  and  the  rest  in  money;  how  much  money  did 
he  pay? 

2.  If  a  barrel  of  flour  is  worth  eight  dollars,  and  a 
hundred  weight  of  sugar  is  worth  twelve  dollars;  how 
much  more  is  the  sugar  worth  than  the  flour? 

3.  If  a  man  had  eleven  dollars,  and  should  buy  three 
bushels  of  corn  for  five  dollars;  how  much  money  would 
he  have  left? 


18  ARITHMETIC.  [part  i. 

4.  A  man  bought  a  jBrkin  of  butter  for  fifteen  dol- 
lars, but  it  being  damaged  he  was  willing  to  sell  it 
again  for  eight  dollars  less  than  he  gave  for  it;  what 
did  he  sell  it  for? 

5.  A  man  bought  three  barrels  of  flour  for  eighteen 
dollars,  and  sold  it  again  for  eleven  dollars;  what  did 
he  lose  by  the  bargain? 

6.  A  man  bought  a  keg  of  tobacco  for  thirteen 
dollars,  and  sold  it  again  for  eighteen;  what  did  he 
gain  by  the  bargain  ? 

7.  Five  less  two  are  how  many  ? 

8.  Seven  less  three  are  how  many? 

9.  Three  less  three  are  how  many? 

10.  Nine  less  three  are  how  many? 

11.  Six  less  two  are  how  many? 

12.  Seven  less  four  are  how  many? 

13.  Eight  less  three  are  how  many? 

14.  Five  less  four  are  how  many? 

15.  Seven  less  five  are  how  many? 

16.  Nine  less  five  are  how  many? 

17.  Eight  less  six  are  how  many? 

18.  Eleven  less  two  are  how  many? 

19.  Twelve  less  four  are  how  many? 

20.  Ten  less  seven  are  how  many  ? 

21.  Thirteen  less  five  are  how  many? 

22.  Fourteen  less  eight  are  how  many? 

23.  Twelve  less  seven  are  how  many? 

24.  Seventeen  less  five  are  how  many? 

25.  Eighteen  less  ten  are  how  many  ? 

26.  Thirteen  less  seven  are  how  many? 

27.  Sixteen  less  seven  are  how  many? 

28.  Fifteen  less  seven  are  how  many? 

29.  Nineteen  less  six  are  how  many? 

30.  Eighteen  less  five  are  how  many? 

31.  Seventeen  less  eight  are  how  many? 

32.  Fourteen  less  nine  are  how  many? 

33.  Sixteen  less  five  are  how  many? 


8EC.  I.]  ARITHMETIC.  19 

3-4.  Fifteen  less  eight  are  how  many? 

35.  Fourteen  less  nine  are  how  many? 

36.  Sixteen  less  ten  are  how  many?  r 
,^37.  Seventeen  less  nine  are  how  many? 

38.  Eighteen  less  seven  are  how  many  ? 

F.  1.  How  many  are  nine  and  two?  Nineteen  and 
tw(J?  Twenty-nine  and  two?  Thirty-nine  and  two? 
Forty-nine  and  two?  Fifty-nine  and  two?  Sixty-nine 
an^^  two?  Seventy-nine  and  two?  Eighty-nine  and 
two?  Ninety-nine  and  two? 

2.  How  many  are  nine  and  three  ?  Nineteen  and 
three?  Twenty-nine  and  three?  Thirty-nine  and 
three?  Forty-nine  and  three?  Fifty-nine  and  three? 
Sixty-nine  and  three  ?  Seventy-nine  and  three  ? 
Eighty-nine  and  three?  Ninety-nine  and  three? 

8.  How  many  are  nine  and  four?  Nineteen  and  four? 
Twenty-nine  and  four?  Thirty-nine  and  four?  Forty- 
nine  and  four?  Fifty-nine  and  four?  Sixty-nine  and 
four?  Seventy-nine  and  four?  Eighty-nine  and  four? 
Ninety-nine  and  four? 

4.  How  many  are  nine  and  five?  Nineteen  and  five? 
Twenty-nine  and  five?  Thirty-nine  and  five?  Forty- 
nine  and  five?  Fifty-nine  and  five?  Sixty-nine  and 
five?  Seventy-nine  and  five?  Eighty-nine  and  five? 
Ninety-nine  and  five? 

5.  How  many  are  nine  and  six?  Nineteen  and  six? 
Twenty-nine  and  six?  Thirty-nine  and  six?  Forty-nine 
and  six?  Fifty-nine  and  six?  Sixty-nine  and  six? 
Seventy-nine  and  six?  Eighty-nine  and  six?  Ninety- 
nine  and  six? 

6.  How  many  are  nine  and  seven?  Nineteen  and 
seven  ?  Twenty-nine  and  seven?  Thirty-nine  and  seven  ? 
Forty-nine  and  seven?  Fifty-nine  and  seven?  Sixty- 
nine  and  seven  ?  Seventy-nine  and  seven  ?  Eighty-nine 
and  seven?  Ninety-nine  and  seven? 

7.  How  many  are  nine  and  eight?  Nineteen  and 
eight?  Twenty-nine  iind  eight?  Thirty -nine  and  eight? 


20  ARITHMETIC.  [part  i. 

Forty-nine  and  eight?  Fifty-nine  and  eight?  Sixty- 
nine  and  eight?  Seventy-nine  and  eight?  Eighty-nine 
and  eight?  Ninety-nine  and  eight? 

8.  How  many  are  nine  and  nine?  Nineteen  a^d 
nine?  Twenty-nine  and  nine?  Thirty-nine  and  nine? 
Forty-nine  and  nine?  Fifty-nine  and  nine?  Sixty-nyie 
and  nine?  Seventy-nine  and  nine?  Eighty-nine  and 
nine?  Ninety-nine  and  nine? 

9.  How  many  are  nine  and  ten?  Nineteen  and  ten? 
Twenty-nine  and  ten  ?  Thirty-nine  and  ten  ?  Forty-n4ne 
and  ten?  Fifty-nine  and  ten?  Sixty-nine  and  ten? 
Seventy-nine  and  ten?  Eighty-nine  and  ten?  Ninety- 
nine  and  ten  ? 

.  10.  How  many  are  eight  and  three?  Eighteen  and 
three?  Twenty-eight  and  three?  Thirty-eight  and 
three?  Forty-eight  and  three?  Fifty-eight  and  three? 
Sixty-eight  and  three?  Seventy-eight  and  three? 
Eighty-eight  and  three  ?  Ninety-eight  and  three  ? 

11.  How  many  are  eight  and  four?  Eighteen  and 
four?  Twenty-eight  and  four?  Thirty-eight  and  four? 
Forty-eight  and  four  ?  Fifty-eight  and  four  ?  Sixty- 
eight  and  four?  Seventy-eight  and  four  ?  Eigkty-eight 
and  four  ?  Ninety-eight  and  four. 

12.  How  many  are  eight  and  five  ?  Eighteen  and 
five?  Twenty-eight  and  five?  Thirty-eight  and  five? 
Forty-eight  and  five  ?  Fifty-eight  and  five?  Sixty- 
eight  and  five  ?  Seventy-eight  and  five  ?  Eighty-eight 
and  five  ?  Ninety-eight  and  five  ? 

13.  How  many  are  eight  and  six  ?  Eighteen  and 
six?  Twenty-eight  and  six?  Thirty-eight  and  six? 
Forty  eight  and  six  ?  Fifty-eight  and  six  ?  Sixty-eight 
and  six?  Seventy-eight  and  six?  Eighty-eight  and 
six?  Ninety-eight  and  six? 

14.  How  many  are  eight  and  seven?  Eighteen  and 
seven?  Twenty-eight  and  seven?  Thirty-eight  and 
seven?  Forty-eight  and  seven?  Fifty-eight  and  seven ? 
Sixty-eight  and  seven?  Seventy-eight  and  seven? 
EightyTcight  and  seven  ?  Ninety-eight  and  seven  ? 


SEC.  I  ]  ARITHMETIC.  21 

15.  How  many  are  eight  and  eiglit?  Eighteen  and 
eio'ht?  Twenty-eight  and  eight i*  Thirty-eight  and 
eight?  Forty-eight  and  eight?  Fifty-eight  and  eight? 
Sixty-eight  and  eight?  Seventy -eight  and  eight? 
Ninety-eight  and  eight? 

16.  How  many  are  eight  and  nine?  Eighteen  and 
nine?  Twenty-eight  and  nine?  Thirty-eight  and  nine? 
Forty-eight  and^nine?  Fifty-eight  and  nine?  Sixty- 
eight  and  nine  ?  Seventy-eight  and  nine?  Eighty-eight 
and  nine?  Ninety  eight  and  nine? 

17.  How  many  arc  seven  and  four?  Seventeen  and 
four?  Twenty  seven  and  four?  Thirty-seven  and  four? 
Forty-seven  and  four?  Fifty-seven  and  four?  Sixty- 
seven  and  four?  Seventy-seven  and  four?  Eighty  seven 
and  four?  Ninety-seven  and  four? 

18.  How  many  are  seven  and  five?  Seventeen  and 
five?  Twenty-seven  and  five?  Thirty-seven  and  five? 
Forty-seven  and  five?  Fifty-seven  and  five?  Sixty- 
seven  and  five?  Seventy-seven  and  five?  Eighty-seven 
and  five?  Ninety-seven  and  five? 

19.  How  many  are  seven  and  six?  Seventeen  and 
six?  Twenty-seven  and  six?  Thirty-seven  and  six? 
Forty-seven  and  six?  Fifty-seven  and  six?  Sixty-seven 
and  six?  Seventy-seven  and  six?  Eighty-seven  and 
six?  Ninety-seven  and  six? 

20.  How  many  are  seven  and  seven  ?  Seventeen  and 
seven?  Twenty-seven  and  seven?  Thirty-seven  and 
seven  ?  Forty-seven  and  seven  ?  Fifty-seven  and  seven  ? 
Sixty-seven  and  seven?  Seventy-seven  and  seven? 
Eighty-seven  and  seven?  Ninety-seven  and  seven  ? 

21.  How  many  are  seven  and  eight?  Sev'enteen  and 
eight?  Twenty-seven  and  eight?  Thirty-seven  and 
eight?  Forty-seven  and  eight?  Fifty -seven  and  eight? 
Sixty-seven  and  eight?  Seventy-seven  and  eight? 
Eighty-seven  and  eight?  Ninety  seven  and  eight? 

22.  How  many  are  seven  and  nine?  Seventeen  and 
nine?  Twenty-seven  and- nine?  Thirty-seven  and  nine ? 
Forty-seven  and  nine?  '^ifty-seven  and  nine?  Sixty- 


22  ARITHMETIC.  [part  i. 

seven   and  nine?    Seventy-seven   and  nine?    Eighty- 
seven  and  nine?  Ninety-seven  and  nine? 

23.  How  many  are  six  and  five?  Sixteen  and  five? 
Twenty-six  and  five?  Thirty-six  and  five?  Forty-six 
and  five?  Fifty-six  and  five?  Sixty-six  and  five? 
Seventy -six  and  five?  Eighty-six  and  five?  Ninety-six 
and  five  ? 

24.  How  many  are  six  and  six?  Sixteen  and  six? 
Twenty-six  and  six?  Thirty-six  and  six?  Forty-six  and 
six?  Fifty-six  and  six?  Sixty-six  and  six  ?  Seventy-six 
and  six?  Eighty -six  and  six?  Ninety -six  and  six? 

25.  How  many  are  six  and  seven?  Sixteen  and 
seven?  Twenty-six  and  seven?  Thirty-six  and  seven? 
Forty-six  and  seven?  Fifty-six  and  seven?  Sixty-six 
and  seven?  Seventy -six  and  seven?  Eighty-six  and 
seven  ?  Ninety-six  and  seven? 

26.  How  many  are  six  and  eight?  Sixteen  and 
eight?  Twenty-six  and  eight?  Thirty-six  and  eight? 
Forty-six  and  eight?  Fifty-six  and  eight?  Sixty -six 
and  eight?  Seventy-six  and  eight?  Eighty-six  and 
eight?    Ninety-six  and  eight? 

27.  How  many  are  six  and  nine?  Sixteen  and  nine  ? 
Twenty-six  and  nine?  Thirty-six  and  nine?  Forty-six 
and  nine?  Fifty -six  an'd  nine?  Sixty-six  and  nine? 
Seventy-six  and  nine?  Eighty-six  and  nine?  Ninety- 
six  and  nine? 

28.  How  many  are  five  and  six?  Fifteen  and  six? 
Twenty-five  and  six?  Thirty-five  and  six?  Forty-five 
and  six?  Fifty-five  and  six?  Sixty-five  and  six? 
Seventy-five  and  six?  Eighty-five  and  six ?  Ninety-five 
and  six?    ''"■ 

29.  How  many  are  five  and  seven?  Fifteen  and 
seven  ?  Twenty-five  and  seven  ?  Thirty-five  and  seven  ? 
Forty-five  and  seven?  Fifty-five  and  seven?  Sixty-five 
and  seven?  Seventy-five  and  seven?  Eighty -five  and 
seven?      Ninety-five  and  seven? 

30.  How  many  are  five  and  eight?  Fifteen  and 
eight?   Twenty-five  and  eight?    Thirty-five  and. eight? 


8BC.  I.]       ^  ARITHMETIC.  2S 

Forty-five  and  eight?  Fifty-five  and  eight?  Sixty-five 
and  eight?  Seventy-five  and  eight?  Eighty-five  and 
eight?     Ninety-five  and  eight? 

31.  How  many  are  five  and  nine?  Fifteen  and  nine? 
Twenty-five  and  nine?  Thirty-five  and  nine?  Forty- 
five  and  nine?  Fifty-five  and  nine?  Sixty-five  and  nine? 
Seventy-five  and  nine?  Eighty-five  and  nine?  Ninety- 
five  and  nine? 

32.  How  many  are  four  and  seven  ?  Fourteen  and 
seven?  Twenty-four  and  seven?  Thirty-four  and 
seven  ?  Forty-four  and  seven  ?  Fifty-four  and  seven  ? 
Sixty-four  and  seven  ?  Seventy-four  and  seven  ? 
Eighty-four  and  seven  ?    Ninety-four  and  seven  ? 

33.  How  many  are  four  and  eight?  Fourteen  and 
eight?  Twenty-four  and  eight?  Thirty-four  and  eight? 
Forty-four  and  eight?  Fifty-four  and  eight?  Sixty-four 
and  eight?  Seventy-four  and  eight?  Eighty-four  and 
eight?    Ninety-four  and  eight ? 

34.  How  many  are  four  and  nine?  Fourteen  and 
nine?  Twenty-four  and  nine  ?  Thirty-four  and  nine? 
Forty-four  and  nine  ?  Fifty-four  and  nine  ?  Sixty-four 
and  nine?  Seventy-four  and  nine?  Eighty- four  and 
nine?     Ninety-four  and  nine? 

35.  How  many  are  three  and  eight?  Thirteen  and 
eight?  Twenty-three  and  eight?  Thirty-three  and 
eight?  Forty-three  and  eight?  Fifty-three  and  eight.'* 
Sixty-three  and  eight?  Seventy-three  and  eight? 
Eighty-three  and  eight?     Ninety-three  and  eight? 

36.  How  many  are  three  and  nine?  Thirteen  and 
nine?  Twenty-three  and  nine?  Thirty -three  and  nine? 
Forty-three  and  nine?  Fifty-three  and  nine?  Sixty- 
three  and  nine?  Seventy-three  and  nine?  Eighty- 
three  and  nine?     Ninety-three  and  nine? 

37.  How  many  are  two  and  nine?  Twelve  and 
nine?  Twenty-two  and  nine?  Thirty -two  and  nine? 
Forty-two  and  nine  ?  Fifty-two  and  nine  ?  Sixty-two 
and  nine?  Seventy-two  and  nine?  Eighty-two  and 
nine  ?     Ninety-two  and  nine  ? 


24  ARITHMETIC.  [parti. 

Or.  i.  A  man  bought  a  firkin  of  butter  for  nine 
dollars,  a  keg  of  molasses  for  six  dollars,  and  five 
bushels  of  wheat  for  seven  dollars ;  how  much  did  he 
give  for  the  whole  ? 

2.  A  boy  gave  some  apples  to  his  companions,  to  one 
he  gave  seven,  to  another  six,  and  to  another  eight; 
how  many  did  he  give  to  all  ? 

3.  A  man  bought  a  cow  for  seventeen  dollars,  a  sheep 
for  nine,  and  a  calf  for  seven ;  how  much  did  he  give 
for  the  whole  ? 

4.  A  drover  bought  sheep  as  follows :  of  one  man 
he  bought  twenty-seven,  of  another  eight,  of  another 
ten,  and  of  another  five;  afterwards  he  sold  nine  of 
them ;  how  many  had  he  then  ? 

5.  A  lady  bought  a  comb  for  thirty-seven  cents,  some 
tape  for  eight  cents,  some  pins  for  ten  cents,  some 
needles  for  six  cents,  and  some  thread  for  six  cents ; 
she  gave  seventy-five  cents ;  how  much  change  ought 
she  to  receive  back  ? 

6.  Eight,  and  nine,  and  six,  are  how  many  ? 

7.  Five,  and  seven,  and  three,  are  how  many? 

8.  Four,  and  three,  less  two,  are  how  many  ? 

9.  Seven,  and  five,  less  three,  are  how  many  ? 

10.  Sixteen,  and  nine,  and  three,  are  how  many  ? 

11.  Twenty-three  and  eight  are  how  many  ? 

12.  Twenty-seven  and  five  are  how  many  ? 

13.  Twenty-five,  less  eight,  are  how  many  ? 

14.  Thirty- two  and  seven,  less  nine,  are  how  many  ? 

15.  Thirty-eight,  and  six,  and  four,  less  seven,  are 
how  many  ? 

16.  Forty-four,  and  eight,  and  three,  and  seven,  are 
how  many  ? 

17.  Fifty-two,  and  six,  and  four,  and  fi,ve,  and  three, 
are  how  many  ? 

18.  Fifty-seven,  and  six,  and  three,  and  five,  and 
two,  less  eight,  are  how  many  ? 

19.  Sixty-three,  and  five,  and  four,  and  six,  and  two, 
less  seven,  are  how  many  ? 


8KC.  I.]  ARITHMETIC.  26 

20.  Seventy-five,  and  six,  and  eight,  and  three,  and 
se\en,  and  four,  less  nine,  are  how  many? 

21.  Eighty-three,  and  six,  and  five,  and  two,  and 
seven,  and  nine,  less  four,  are  how  many  ? 

22.  Fifty-eight,  and  ten,  and  five,  and  seven,  and 
three,  and  six,  and  four,  less  nine,  are  how  many  ? 

23.  Sixty-seven,  and  five,  and  eight,  and  nine,  and 
seven,  less  six,  are  how  many? 

24.  Seventy-four,  and  nine,  and  seven,  and  five,  and 
two,  less  six,  are  how  many  ? 

25.  Seventy-eight,  and  seven,  and  six,  and  two,  and 
five,  and  eight,  less  nine,  are  how  many  ? 

26.  Eighty-four,  and  seven,  and  six,  and  eight,  and 
five,  less  ten,  are  how  many? 

27.  Forty-seven,  and  eight,  and  six,  and  two,  and 
four,  and  eight,  and  three,  and  seven,  and  ten,  and  nine, 
less  five,  are  how  many  ? 

28.  Thirty-five,  and  eight,  and  four,  and  ^ix,  and 
three,  and  four,  less  eleven,  are  how  many  ? 

29.  Seventy,  and  ten,  and  six,  and  nine,  and  seven, 
and  two,  and  five,  and  eight,  and  nine,  less  three,  are 
how  many  ? 

H.  1.  A  man  bought  a  cow  for  twenty-eight  dollars, 
and  a  sheep  for  four  dollars,  and  a  pig  for  seven  dollars  ; 
how  much  did  he  give  for  the  whole  ? 

2.  James  had  twenty-seven  cents;  John  gave  him 
four  more,  David  seven,  and  George  eleven,  and  he 
bought  nine  cents  worth  of  cake ;  how  many  cents  had 
he  left? 

3.  A  man  paid  sixteen  dollars  to  A,  nine  dollars  to 
B,  seven  dollars  to  0,  ten  dollars  to  D,  six  dollars  to  E, 
four  dollars  to  F,  and  had  eight  dollars  left ;  how  many 
had  he  at  first  ? 

4.  From  Brown's  to  Black's  it  is  three  miles ;  from 
Black's  to  Blue's,  six  miles ;  from  Blue's  to  Green's, 
eleven  miles ',  from  Green's  to  Violet's,  four  miles ; 
from  Violet's    to    Olive's,  four    miles ;    from    Olive'a 


26  ARITHMETIC.  [part  i. 

to  Purple's,  nine  miles;  from  Purple's  to  White's, 
four  miles;  how  many  miles  is  it  from  Brown's  to 
White's  ? 

5.  One  boy  had  fifteen  nuts;  another  boy  gave  him 
seven;  another,  nine;  and  another  gave  him  enough 
to  make  his  number  forty;  how  many  did  the  last  boy 
give  him  ? 

6.  A  boy  had  thirty-seven  apples ;  he  gave  five  to 
one  companion;  and  eight  to  another;  and  when  he 
had  given  some  to  another,  he  had  six  left;  how  many 
did  he  give  to  the  last  ? 

7.  A  man  owed  fifty-six  dollars  ;  at  one  time  he  paid 
seventeen  dollars;  at  another,  eight;  at  another,  five; 
at  another,  seven;  at  last  he  paid  the  rest  of  the  debt, 
wanting  four  dollars ;  how  uiuch  was  the  last  payment  ? 

8.  Six  men  bought  a  horse  for  seventy  dollars ;  the 
first  gave  twenty-three  dollars ;  the  second,  fifteen ;  the 
third,  twelve ;  the  fourth,  nine ;  the  fifth,  seven ;  how 
much  did  the  sixth  give  ? 

9.  A  man  bought  a  horse  for  forty-five  dollars  and 
paid  fifteen  dollars  for  keeping  him ;  he  let  him  enough 
to  receive  twenty  dollars ;  and  then  sold  him  for  forty- 
three  dollars;  did  he  gain  or  lose  by  the  bargain  ?  and 
how  much  ? 


SECTION  II. 

A.  1.  What  cost  three  yards  of  tape,  at  two  cents  a 
yard  ?  * 

2.  What  cost  four  apples,  at  two  cents  apiece  ?     ' 

3.  What  cost  five  peaches,  at  three  cents  apiece? 

4.  What  must  you  give  for  two  oranges,  at  six  cents 
apiece  ? 

♦The  pupil  should  be  mado  to  observe  that  three  yards  will  cost 
three  times  as  much  as  one  yard ;  and  say,  if  one  yard  costs  two  cents, 
three  yards  will  cost  three  times  two  cents.  He  should  be  made  to 
jfive  this  reason  for  the  solution  of  each  question,  varying  tlie  nunibera 
according  to  the  question. 


SEC.  II.]  ARITHMETIC.  27 

5.  What  would  be  the  price  of  three  barrels  of  cider, 
at  three  dollars  a  barrel  ? 

6.  If  one  orange  is  worth  three  apples,  how  many 
apples  are  four  oranges  worth  ? 

7.  What  are  two  barrels  of  flour  worth,  at  five  dollars 
a  barrel  ? 

8.  What  cost  three  yards  of  cloth,  at  four  dollars  a 
yard  ? 

9.  What  cost  two  pounds  of  raisins,  at  eight  cents  a 
pound  ? 

10.  What  cost  three  lemons,  at  six  cents  apiece  ? 

11.  If  a  man  travels  three  miles  in  an  hour,  how  many 
miles  will  he  travel  in  four  hours  ? 

12.  What  will  five  pair  of  shoes  come  to  at  two  dol- 
lars a  pair  ? 

13.  What  is  the  price  of  seven  yards  of  cloth,  at 
three  dollars  a  yard  ? 

14.  What  is  the  value  of  two  pounds  of  beef,  at 
seven  cents  a  pound  ? 

15.  If  there  are  three  feet  in  one  yard,  how  many 
feet  are  there  in  four  yards? 

16.  How  many  feet  are  there  in  seven  yards  ? 

17.  How  many  feet  are  there  in  six  yards  and  two 
feet  ? 

18.  If  a  man  earns  seven  dollars  in  one  week,  how 
much  would  he  earn  in  five  weeks  ? 

19.  What  cost  seven  hundred  weight  of  sugar,  at  nine 
dollars  a  hundred  weight? 

20.  What  cost  seven  pounds  of  sugar,  at  ten  cents  a 
pound  ? 

21.  If  one  half  yard  of  cloth  costs  three  dollars,  what 
would  three  yards  cost  ? 

22.  If  one  quarter  of  a  yard  of  cloth  costs  two  dol- 
lars, what  is  that  a  yard? 

23.  How  many  yards  of   cloth  are  there  in  seven 
pieces,  each  piece  containing  ten  yards  ? 

24.  What  will  five  barrels  of  flour  cost,  at  six  dol- 
lars a  barrel  ? 


28  ARITHMETIC.  [part  i. 

25.  Jf  a  man  can  travel  four  miles  in  an  hour,  how 

far  can  he  travel  in  eight  hours  ? 

26.  If  it  takes  four  bushels  of  wheat  to  make  a  bar- 
rel of  flour,  how  many  bushels  will  it  take  to  make 
seven  barrels  ? 

B.  1.  Two  times  one  are  how  many?* 
2.  Two  times  two  are  how  many? 
8.  Two  times  three  are  how  many? 
4.  Two  times  four  are  how  many  ? 
6.  Two  times  five  are  how  many  ? 

6.  Two  times  six  are  how  many  ? 

7.  Two  times  seven  are  how  many  ? 

8.  Two  times  eight  are  how  many  ? 

9.  Two  times  nine  are  how  man}'  ? 

10.  Two  times  ten  are  how  many  ? 

11.  Three  times  one  are  how  many? 

12.  Three  times  two  are  how  many  ? 

13.  Three  times  three  are  how  many? 

14.  Three  times  four  are  how  many  ? 

15.  Three  times  five  are  how  many  ? 

16.  Three  time's  six  are  how  many  ? 

17.  Three  times  seven  are  how  many? 

18.  Three  times  eight  are  how  many  ? 

19.  Three  times  nine  are  how  many? 

20.  Three  times  ten  are  how  many? 

21.  Four  times  one  are  how  many  ? 

22.  Four  times  two  are  how  many  ? 

23.  Four  times  three  are  how  many? 

24.  Four  times  four  are  how  many? 

25.  Four  times  five  are  how  many  ? 

26.  Four  times  six  are  how  many  ?  . 

27.  Four  times  seven  are  how  many  ? 

28.  Four  times  eight  are  how  many? 

29.  Four  times  nine  are  how  many  ? 
80.  Four  times  ten  are  how  many  ? 

*  See  the  Key. 


SEC.  ii.J  ARITHMETIC.  29 

•       31.  Five  times  one  are  bow  many  ? 

32.  Five  times  two  are  how  many  ? 

33.  Five  times  three  are  how  many  ? 

34.  Five  times  four  are  how  many  ? 

35.  Five  times  five  are  how  many  ? 

36.  Five  times  six  are  how  many? 

37.  Five  times  seven  are  how  many  ? 

38.  Five  times  eight  are  how  many  ? 

39.  Five  times  nine  are"  how  many  ? 

40.  Five  times  ten  are  how  many  't 

41.  Six  times  one  are  how  many  ? 

42.  Six  times  two  arc  how  many  ? 

43.  Six  times  three  are  how  many  ? 

44.  Six  times  four  are  how  many  ? 

45.  Six  times  five  are  how  many  ? 

46.  Six  times  six  are  how  many  ? 

47.  Six  times  seven  are  how  many  ? 

48.  Six  times  eight  are  how  many  ? 

49.  Six  times  nine  are  how  many? 

50.  Six  times  ten  are  how  many  ? 

51.  Seven  times  one  are  how  many  ? 

52.  Seven  times  two  are  how  many  ? 

53.  Seven  times  three  are  how  many? 

54.  Seven  times  four  are  how  many  ? 

55.  Seven  times  five  are  how  many  ? 

56.  Seven  times  six  are  how  many  ? 

57.  Seven  times  seven  are  how  many  ? 

58.  Seven  times  eight  are  how  many  ? 

59.  Seven  times  nine  are  how  many  ? 

60.  Seven  times  ten  are  how  many  ?  • 

61.  Eight  times  one  are  how  many? 

62.  Eight  times  two  are  how  many  ? 

63.  Eight  times  three  arc  how  many? 

64.  Eight  times  four  are  how  many? 

65.  Eight  times  five  are  how  many  ? 

66.  Eight  times  six  are  how  many? 

67.  Eight  times  seven  are  how  many  ? 

68.  Eight  times  eight  are  how  many  ? 


80  ARITHMETIC.  [pakt'i. 

69.  Eight  times  nine  are  how  many  ?  g 

70.  Eight  times  ten  are  how  many  ? 

71.  Nine  times  one  are  how  many?  . 

72.  Nine  times  two  are  how  many?  < 

73.  Nine  times  three  are  how  many?  ; 

74.  Nine  times  four  are  how  many?  : 

75.  Nine  times  five  are  how  many? 
7G.  Nine  times  six  are  how  many? 

77.  Nine  times  seven  are  how  many? 

78.  Nine  times  eight  are  how  many?  : 

79.  Nine  times  nine  are  how  many? 

80.  Nine  times  ten  are  how  many? 

81.  Ten  times  one  are  how  many? 

82.  Ten  times  two  are  how  many? 

83.  Ten  times  three  are  how  many? 

84.  Ten  times  four  are  how  many? 

85.  Ten  times  five  are  how  many? 

86.  Ten  times  six  are  how  many? 

87.  Ten  times  seven  are  how  many? 

88.  Ten  times  eight  are  how  many? 

89.  Ten  times  nine  are  how  many? 

90.  Ten  times  ten  are  how  many? 

C.   1.  Two  times  two  are  how  many  times  one?     "'• 

2.  Three  times  two  are  how  many  times  one? 

3.  Four  times  two  are  how  many  times  one?     '* 

4.  Five  times  two  are  how  many  times  one? 

5.  Seven  times  two  are  how  many? 

6.  Nine  times  two  are  how  many  ? 

7.  Six  times  two  are  how  many? 

8.  Eight  times  two  are  how  many? 

9.  Ten  times  two  are  how  many? 

10.  Two  times  three  are  how  many? 

11.  Three  times  three  are  how  many? 

12.  Four  times  three  are  how  many?  • 

13.  Five  times  three  are  how  many? 

14.  Six  times  three  are  how  many? 

15.  Eight  times  three  are  how  many? 


SEC.  II.]  ARITHMETIC.  81 

16.  Seven  times  three  are  how  many? 

17.  Ten  times  three  are  how  many? 

18.  Nine  times  three  are  how  many? 

19.  Two  times  four  are  how  many? 

20.  Six  times  four  are  how  many? 

21.  Four  times  four  are  how  many? 

22.  Seven  times  four  are  how  many? 
«     28,  Nine  times  four  are  how  many? 

24.  Three  times  four  are  how  many? 

25.  Five  times  four  are  how  many? 

26.  Ten  times  four  are  how  many? 

27.  .Eight  times  four  are  how  many? 

28.  Two  times  five  are  how  many? 

29.  Five  times  five  are  how  many? 

30.  Three  times  five  are  how  many? 

31.  Six  times  five  are  how  many? 

32.  Two  times  six  are  how  many? 

33.  Four  times  five  are  how  many? 

34.  Seven  times  five  are  how  many? 

35.  Three  times  six  are  how  many? 

36.  Seven  times  six  are  how  many? 

37.  Seven  times  seven  are  how  many? 

38.  Four  times  eight  are  how  many? 
39,.  Six  times  seven  are  how  many? 

40.  Eight  times  nine  are  how  many? 

41.  Six  times  eight  are  how  many? 

42.  Three  times  seven  are  how  many? 

43.  Four  times  nine  are  how  many? 

44.  Three  times  eight  are  how  many? 

45.  Six  times  six  are  how  many? 

46.  Six  times  nine  are  how  many? 

47.  Nine  times  five  arc  how  many? 

48.  Four  times  six  are  how  many? 

49.  Two  times  nine  are  how  many? 

50.  Seven  times  nine  are  how  many? 

51.  Nine  times  eight  are  how  many? 
62.  Two  times  eight  are  how  many? 
53.  Three  times  ten  are  how  many? 


82  ARITHMETIC.  [pabt  i. 

64.  Eight  times  seven  are  how  many? 

55.  Five  times  six  are  how  many? 

56.  Five  times  eight  arc  how  many? 

57.  Two  times  seven  are  how  many? 

58.  Two  times  six  arc  how  many? 

59.  Eight  times  six  are  how  many? 

60.  Four  times  seven  are  how  many? 

61.  Eight  times  eight  are  how  many?  * 

62.  Ten  times  five  are  how  many? 

63.  Seven  times  ten  arc  how  many? 

64.  Ten  times  ten  are  how  many? 

65.  Nine  times  six  are  how  many? 

66.  Five  times  nine  are  how  many? 

67.  Three  times  nine  are  how  many  ? 

68.  Nine  times  seven  are  how  many? 

69.  Five  times  ten  are  how  many? 

70.  Seven  times  eight  are  how  many? 

71.  Five  times  seven  are  how  many? 

72.  Ten  times  eight  are  how  many  ? 

73.  Ten  times  seven  are  how  many? 

74.  Nine  times  ten  are  how  many? 

75.  Eight  times  five  are  how  many? 

76.  Nine  times  nine  are  how  many? 

77.  Four  times  ten  are  how  many?      , 

78.  Ten  times  six  are  how  many? 

79.  Eight  times  ten  are  how  many? 

80.  Ten  times  nine  are  how  many? 

D.  1.  What  cost  three  yards  of  cloth,  at  five  dollars 
a  yard  ? 

2.  What  cost  four  oranges,  at  six  cents  apiece? 

8.  What  cost  seven  barrels  of  cider,  at  three  dollars 
a  barrel  ? 

4.  How  much  do  three  barrels  of  beer  come  to,  at 
seven  dollars  a  barrel? 

5.  What  cost  four  firkins  of  butter,  at  eight  dollars 
a  firkin  ? 

6.  What  do  nine  pounds  of  veal  come  to,  at  six 
cents  a  pound  ? 


8E0.  II.]  ARITHMETIC.  83 

7.  What  cost  six  reams  of  paper,  at  five  dollars  per 
ream  ? 

S.  What  cost  eight  pairs  of  shoes,  at  three  dollars  a 
pairi* 

9.  What  is  the  value  of  nine  yards  of  cloth,  at  six 
dollars  a  yard  ? 

10.  If  a  man  travels  five  miles  an  hour,  how  many 
miles  will  he  travel  in  nine  hours? 

11.  There  is  an  orchard  consistini;  of  ten  rows  of 
trees,  and  nine  trees  in  each  row;  how  many  trees  are 
there  in  the  orchard? 

12.  On  a  chess-board  there  are  eight  rows  of  squares, 
and  eight  squares  in  each  row;  how  many  squares  are 
there  on  the  board  ? 

13.  In  one  penny  there  are  four  farthings;  how 
many  farthings  are  there  in  six  pence? 

1-1.   How  many  farthings  are  there  in  eight  pence? 

15.  IIow  many  farthings  are  there  in  nine  pence  ? 

16.  How  many  farthings  are  there  in  ten  pence? 

17.  In  one  siiilling  there  are  twelve  pence;  how 
many  farthings  are  there  in  a  shilling? 

IS.  In  one  pint,  there  are  four  gills;  how  many  gills 
are  there  in  five  pints? 

19.  In  one  quart,  there  are  two  pints;  how  many 
pints  are  there  in  six  quarts? 

20.  How  many  pints  are  there  in  three  quarts? 

21.  How  many  gills  are  there  in  six  pints? 

22.  How  many  gills  are  there  in  one  quart? 

23.  How  many  gills  are  there  in  three  quarts? 

24.  In  one  gallon  there  are  four  quarts;  how  many 
quarts  are  there  in  three  gallons  ? 

25.  How  many  quarts  are  there  in  five  gallons? 

26.  How  many  quarts  are  there  in  seven  gallons  ? 

27.  How  many  pints  are  there  in  one  gallon? 

28.  How  many  pints  are  there  in  three  gallons? 

29.  How  many  gills  are  there  in  one  gallon  ? 

30.  How  many  gills  are  there  in  five  quarts  ? 

31.  How  many  gills  are  there  in  two  gallons  ? 

2 


84  ARITHMETIC.  [pakt  i. 

32.  A  person  bought  two  oranges,  at  six  cents 
apiece;  and  seven  lemons,  at  four  cents  apiece;  and 
five  pears,  at  two  cents  apiece;  how  much  did  the 
whole  come  to  ? 

33.  If  one  pint  of  oil  costs  eight  cents;  what  will  one 
quart  cost? 

3-4.  If  one  gill  of  alcohol  costs  four  cents,  what  will 
one  quart  cost? 

35.  If  one  gill  of  beer  costs  two  cents,  what  will  one 
gallon  cost  ? 

36.  If  a  stage  runs  seven  miles  in  an  hour^  how  far 
will  it  run  in  nine  hours? 

37.  Two  men  start  from  the  same  place  and  travel 
different  ways ;  one  travels  two  miles  in  an  hour ;  the 
other  travels  three  miles  in  an  hour ;  how  far  apart 
will  they  be  at  the  end  of  one  hour  ?  How  far  at  the 
end  of  two  hours?  How  far  at  the  end  of  three 
hours  ?      How  far  at  the  end  of  four  hours? 

38.  Two  men  start  from  the  same  place  and  travel 
the  same  way;  one  travels  at  the  rate  of  two  miles 
in  an  hour;  the  other,  four;  how  far  apart  will  they 
be  in  one  hour?  How  far  in  two  hours  ?  How  far  in 
four  hours? 

39.  If  three  men  can  do  a  piece  of  work  in  two 
days,  how  many  days  would  it  take  one  man  to  do  it? 

40.  If  four  men  can  do  a  piece  of  work  in  five  days, 
how  many  days  would  it  take  one  man  to  do  it  ? 

41.  If  six  men  can  do  a  piece  of  work  in  seven 
days,  how  many  men  would  it  take  to  do  it  in  one 
day? 

42.  If  a  quantity  of  provisions  will  serve  three  men 
five  days,  how  many  men  would  it  serve  one  day  ? 

43.  If  a  quantity  of  provisions  will  serve  five  men 
seven  days,  how  many  days  would  it  serve  one  man  ? 

44.  If  fifteen  dollars'  worth  of  provisions  will  serve 
eight  men  five  days,  how  many  days  will  it  serve  one 
man  ? 

45.  A  man  had  a  piece  of  work  to  perform  which 


SEC.  III.]  ARITHMETIC.  35 

seven  men  could  do  in  nine  days,  but  it  was  necessary 
that  the  whole  should  be  completed  in  one  day;  how 
many  men  must  he  employ? 

4G.  If  the  interest  of  one  dollar  is  six  cents  a  year, 
what  would  be  the  interest  of  tea  dollars  for  the  same 
time  ? 

47.  If  the  interest  of  one  dollar  is  six  cents  for  one 
year,  what  would  be  the  interest  of  it  for  two  years? 
for  three  years?  for  seven  years? 

48.  If  a  man  can  earn  seven  shillings  in  a  day,  how 
many  shillings  will  he  earn  in  six  days? 

49.  If  a  man  can  earn  eight  dollars  in  a  month,  how 
much  can  he  earn  in  six  months? 

50.  At  five  dollars  a  week,  what  will  nine  weeks' 
board  come  to? 

51.  A  lady  bought  three  yards  of  cambric  at  two 
dollars  a  yard,  seven  yards  of  silk  for  three  dollars  a 
yard,  five  yards  of  ribbon  for  four  dollars,  and  some 
crape  for  two  dollars;  she  paid  four  ten-dollar  bills; 
how  much  must  she  receive  back  again  ? 

SECTION  III. 

A.  1.  How  many  apples,  at  one  cent  apiece,  can 
you  buy  for  four  cents? 

2.  How  many  pears,  at  two  cents  apiece,  can  you 
buy  for  four  cents? 

o.  How  many  peaches,  at  three  cents  apiece,  can 
you  buy  for  six  cents  ? 

4.  How  many  apples,  at  two  cents  apiece,  can  you 
buy  for  six  cents?     How  many  for  eight  cents? 

5.  How  many  pears,  at  three  cents  apiece,  can  you 
buy  for  nine  cents  ?     How  many  for  twelve  cents  ? 

6.  If  you  have  eight  apples  to  give  to  four  boys, 
how  many  can  you  give  to  each  ? 

7  If  a  man  travels  six  miles  in  two  hours,  how  many 
miles  does  he  travel  in  an  hour? 

8.  If  a  man  travels  three  miles  in  an  hour,  how  many 
bours  will  it  take  him  to  travel  nine  miles  ? 


36  ARITHMETIC.  [part  i. 

9.  How  many  yards  of  cloth,  at  three  dollars  a  yard, 
can  you  buy  for  fifteen  dollars  ? 

10.  If  you  had  sixteen  cents,  how  many  cakes  could 
you  buy  at  four  cents  apiece  ? 

11.  If  you  had  ten  dollars,  how  much  cloth  could 
you  buy  at  five  dollars  a  yard  ? 

12.  If  you  had  twelve  apples  to  give  to  six  of  your 
companions,  how  many  would  you  give  them  apiece  ? 

13.  If  a  man  can  travel  six  miles  in  an  hour,  how 
long  will  it  take  him  to  travel  eighteen  miles  ? 

14.  If  a  man  can  travel  five  miles  in  an  hour,  how 
long  will  it  take  him  to  travel  twenty  miles  ? 

15.  In  a  certain  orchard  there  are  twenty-four  trees 
standing  in  rows,  and  there  are  three  trees  in  each 
row  ;  how  many  rows  are  there  ? 

16.  In  an  orchard  there  are  twenty-one  trees,  and 
there  are  seven  trees  in  each  row;  how  many  rows  are 
there  ? 

17.  A  man  paid  twenty-seven  dollars  for  some  sheep, 
and  he  gave  nine  dollars  apiece  for  them;  how  many 
sheep  did  he  buy? 

18.  A  man  paid  twenty -eight  dollars  for  seven  bar- 
rels of  cider;  how  much  did  he  give  a  barrel  ? 

19.  At  five  ceuts  a  piece,  how  many  oranges  can 
you  buy  for  thirty  cents? 

20.  Twenty-five  are  how  maYiy  times  five  ? 

21.  Thirty-two  are  how  many  times  four  ?  How 
many  times  eight  ? 

22.  Thirty-five  are  how  many  times  seven  ?  How 
many  times  five? 

23.  Thirty-six  are  how  many  times  six?  How  many 
times  nine  ?     How  many  times  four  ?  • 

B.  Remarh.  When  any  thing,  or  any  number,  is 
divided  into  two  equal  parts,  one  of  the  parts  is  called 
the  half  of  the  thing  or  the  number. 

1.  If  an  apple  is  worth  two  cents,  what  is  one  half 
of  it  worth  ? 


SKC.  Ill]  ARlTiniETIC.  '37 

2.  AVhat  is  one  half  .of  two  cents? 
Ans.  One  cent. 

^Q.  Why? 

A.  Because,  if  you  divide  two  cents  into  two  equal 
parts,  one  of  the  parts  is  one  oent. 

3.  If  you  can  buy  a  cake  for  two  cents,  how  much 
can  you  buy  for  one  cent  ? 

4.  One  is  whht  part  of  two? 
Ans.  One  is  the  half  part  of  two. 

5.  Two  times  one  are  how  many  times  two? 

G.  If  you  can  buy  one  pear  for  two  cents,  how  many 
can  you  buy  for  three  cents? 

7.  Three  are  how  many  times  two? 
Ans.  Once  two  and"  one  kalf  of  two. 

8.  Four  arc  how  many  times  two? 

9.  If  two  shillinfTS  will  buy  one  yard  of  cotton  cloth, 
how  many  yards  will  five  shillings  buy  ? 

10.  Five  are  how  many  times  two  ? 
Ans,  Two  times  two  and  half  of  two. 

11.  Six  are  how  many  times  two? 

12.  If  two  dollars  will  buy  a  yard  of  cloth,  how 
many  yards  will  seven  dollars  buy  ? 

lo.  How  many  halves  make  a  whole  one? 

14.  Eight  arc  how  many  times  two  ? 

15.  Nine  arc  how  many  times  two? 

16.  Ten  are  how  many  times  two? 

Remarks  When  any  thing,  or  any  number,  is 
divided  into  three  equal  parts,  one  of  those  parts  is 
called  the  third  part  of  the  thing  or  number.  When 
it  is  divided  into  four  equal  parts,  one  part  is  called  the 
fourth  part,  and  so  on. 

17.  If  a  yard  of  cloth  be  worth  three  dollars,  and 
it  be  cut  into  three  equal  pieces,  what  will  one  of  the 
pieces  be  worth  ?  that  is,  what  will  one  third  of  a  yard 
be  worth  I 

18.  What  is  a  third  of  three? 

19.  Suppose  the  yard  of  cloth  to  be  cut  as  before, 
what  will  two  pieces  of  it  cost  ?  that  is,  what  will  two 
thirds  of  a  yard  cost  ? 


38  ARITHMETIC.  [part  i. 

20.  What  is  two  thirds  of  three  ? 

21.  If  three  shillings  will  buy  one  bushel  of  corn, 
what  part  of  a  bushel  will  one  shilling  buy?  What 
part  of  a  bushel  will  two  shillings  buy? 

22.  One  is  what  part  of  three  ? 

Ans.  One  is  the  third  part  of  three;  or  one  third 
of  three. 

23.  Two  is  what  part  of  tlu'ee  ? 

A71S.  Two  is  two  times  the  third  part  of  three,  or 
two  thirds  of  three.* 

24.  Three  times  one  are  how  many  times  three  ? 

25.  If  you  can  buy  a  barrel  of  cider  for  three  dol- 
lars, how  much  can  you  buy  for  four  dollars  ?  How 
much  for  five  dollars  ? 

26.  How  many  thirds  make  a  whole  one? 

27.  Five  are  how  many  times  three  ? 
Ans.  One  three,  and  two  thirds  of  three. 

28.  Six  are  how  many  times  three  ? 

29.  If  you  can  buy  a  barrel  of  fish  for  three  dollars, 
how  much  can  you  buy  for  seven  dollars  ?  How  much 
for  eight  dollars  ? 

30.  What  do  you  understand  by  a  third,  and  by 
two  thirds  of  any  thing  ? 

For  the  answer,  see  remark  after  example  16th. 

31.  Eight  are  how  many  times  three? 

32.  Nine  are  how  many  times  three  ? 

33.  Ten  are  how  many  times  three? 

34.  Eleven  are  how  many  times  three? 

35.  Twelve  are  how  many  times  three? 

36.  If  a  yard  of  cloth  be  worth  four  dollars,  and  it 
be  cut  into  four  equal  parts,  what  will  one  of  the  parts 
be  worth?  that  is,  what  is  one  fourth  of  it  worth? 
What  are  two  fourths  of  it  worth?  What  are  three 
fourths  of  it  worth? 

37.  If  you  can  buy  a  barrel  of  cider  for  four  dol- 

*  The  instructor  may  use  either  ofthe.se  expressions;  the  former 
will  be  most  easily  understood  by  the  pupil :  it  would  be  well,  there- 
fore, for  the  instructor  to  use  tJie  "former  frequently,  though  the  latter 
is  used  for  the  most  part  in  this  treatise. 


SEC.  III.]  ARITHMETIC.  39 

lars,  how  mucli  can  you  buy  for  one   dollar?     How 
much  for  two  dollars?     How  much  for  three  dollars? 

38.  What  part  of  four  is  one? 
Ans.  One  is  the  fourth  part  of  fgur. 

39.  What  part  of  four  is  two  ? 
Alls.  Two  fourths  of  four. 

40.  What  part  of  four  is  three? 
Ans.  Three  fourths  of  four. 

41.  How  many  fourths  make  a  whole  one? 

42.  If  you  can  buy  a  bushel  of  corn  for  four  shil- 
lings, how  much  can  you  buy  for  five  shillings? 
How  much  for  six  shillings?  How  much  for  seven 
shillings  ? 

43.  Five  are  how  many  times  four  ? 
Ans.  Once  four,  and  one  fourth  of  four. 

44.  Six  are  how  many  times  four  ? 

Ans.  Once  four,  and  two  fourths  of  four. 

45.  Seven  are  how  many  times  four? 
Ans.  Once  four,  and  three  fourths  of  four. 

46.  Eight  are  how  many  times  four  ? 

47.  If  four  bushels  of  corn  will  buy  one  yard  of 
cloth,  how  many  yards  will  nine  bushels  buy  ?  How 
many  yards  will  ten  bushels  buy  ?  How  many  yards 
will  eleven  bushels  buy  ? 

48.  What  do  you  understand  by  one  fourth,  two 
fourths,  or  three  fourths  of  any  thing  ? 

See  remark  after  example  16th. 

49.  Ten  are  how  many  times  four  ? 

50.  Eleven  are  how  many  times  four  ? 

51.  Twelve  are  how  many  times  four? 

52.  -Thirteen  are  how  many  times  four  ? 

53.  Fourteen  are  how  many  times  four? 

54.  Fifteen  are  how  many  times  four  ? 

55.  Sixteen  are  how  many  times  four  ? 

56.  If  a  barrel  of  flour  be  worth  five  dollars,  and  it 
be  divided  equally  among  five  men,  what  will  one  man's 
share  be  worth  ?  that  is,  what  is  one.  fifth  of  a  barrel 
worth  ?     What  are  two  fifths  of  it  worth  ?     What  are 


40  ARITHMETIC.  [part  t. 

three  fifths  of  it  worth  ?     What  are  four  fifths  of  it 
worth? 

57.  If  five  dollars  will  buy  one  box  of  butter,  what 
part  of  a  box  will  one  dollar  buy  ?  What  part  will  two 
dollars  buy  ?  What  part  will  three  dollars  buy  ?  What 
part  will  four  dollars  buy  ? 

58.  What  part  of  five  is  one  ? 
Alls.  One  is  the  fifth  part  of  five. 

59.  Two  is  what  part  of  five  ? 

A71S.  Two  fifths  of  five.  * 

60.  Three  is  what  part  of  five  ? 
Ans.  Three  fifths  of  five. 

61.  Four  is  what  part  of  five  ? 

6'2.  How  many  fifths  make  a  whole  one  ? 

63.  If  cherries  are  five  cents  a  quart,  how  many  quarts 
can  you  buy  for  six  cents  ?  How  many  for  seveu  cents  ? 
How  many  for  eight  cents  ?  How  many  for  nine  cents  ?• 
How  many  for  eleven  cents  ?  How  many  for  thirteen 
cents  ? 

64.  What  do  you  understand  by  one  fifth,  two  fifths, 
etc.,  of  any  thing? 

See  remark  after  example  16th. 

65.  Seven  are  how  many  times  five  ? 
A71S.   Once  five  and  two  fifths  of  five. 

66.  Eight  are  how  many  times  five  ? 

67.  Nine  are  how  many  times  five? 

68.  Ten  are  how  many  times  five  ?  • 

69.  Eleven  are  how  many  times  five  ? 

70.  Twelve  are  how  many  times  five  'f 

71.  Thirteen  are  how  many  times  five  ? 

72.  Fourteen  are  how  many  times  five  ? 
78..  Fifteen  are  how  many  times  five  ? 

74.  If  a  barrel  of  beef  cost  six  dollars,  and  it  were 
divided  into  six  equal  parts,  what  would  one  of  the 
parts  be  worth  ?  that  is,  what  is  one  sixth  of  it  worth  ? 
What  are  two  sixths  of  it  worth  ?  What  are  three 
sixths  of  it  worth?     Four  sixths?     Five  sixths? 

75.  If  fish  is  Worth  six  dollars  a  barrel,  what  part  of 


SEC.  III.]  ARITHMETIC.  41 

a  barrel  will  one  dollar  buy  ?  What  part  of  a  barrel 
will  two  dollars  buy?  Three  dollars?  Four  dollars? 
Five  dollars  ? 

76.  What  part  of  six  is  one  ? 
Ans.   One  is  the  sixth  part  of  six. 

77.  Wliat  part  of  six  is  two  ? 
Ans.  Two  sixths  of  six. 

78.  Three  is  what  part  of  six  ? 
Ans.  Three  sixths  of  six. 

79.  Four  is  what  part  of  six? 

80.  How  many  sixths  make  a  whole  one  ? 

81.  How  mucli  rye  at  six  shillings  a  bushel  can  you 
buy  for  sevea  shillings  ?  How  much  for  eight  shil- 
lings ?  Nine  shillings  ?  Ten  shillings  ?  Eleven  shil- 
lings ?  Twelve  shillings?  Thirteen  shillings?  Fif- 
teen shillings?     Seventeen  shillings? 

82.  What  do  j-ou  understand  by  one  sixth,  two  sixths, 
etc.? 

83.  Eight  are  how  many  times  six  ? 
Ans.   One  time  six  and  tw(3  sixths  of  six  ? 

84.  Nine  are  how  many  times  six  ? 

85.  Ten  are  how  many  times  six  ? 

86.  Eleven  are  how  many  tijues  six  ? 

87.  Twelve  are  how  many  times  six? 

88.  Thirteen  are  how  many  times  six  ? 

89.  Fourteen  arc  how  many  times  six  ? 
00.   Fifteen  are  how  many  times  six  ? 

91.  If  coal  is  wqi'th  seven  dollars  a  chaldron,  what 
is  one  seventh  of  a  chaldron  worth?  What  are  two 
sevenths  of  a  chaldron  worth  ?  Three  sevenths  ?  Four 
sevenths  ?     Five  sevenths'?     Six  sevenths  ? 

92.  At  the  rate  of  seven  dollars  a  yard,  how  much 
broadcloth  can  you  buy  for  one  dollar  ?  How  much  for 
two  dollars  ?  How  much  for  three  dollars  ?  How  much 
for  four  dollars  ?  How  much  for  five  dollars  ?  How 
much  for  six  dollars?  How  much  for  eight  dollars? 
How  much  for  ten  dollars  ?  How  much  for  twelve  dol- 
lars ?     How  much  for  fifteen  dollars  ? 


42  ARITHMETIC.  [parti. 

93.  "What  part  of  seven  is  one  ? 
Ans.  One  is  one  seventh  of  seven. 

94.  What  part  of  seven  is  two? 
Ans.  Two  se-venths  of  seven. . 

95.  What  part  of  seven  is  three? 

96.  Four  is  what  part  of  seven? 

97.  Five  is  what  part  of  seven  ? 

98.  What  do  you  understand  by  one  seventh,  two 
sevenths,  etc.,  of  any  thing  ? 

99.  How  many  sevenths  make  a  whole  one  ? 

100.  Nine  are  how  many  times  seyen  ? 

101.  Ten  are  how  many  times  seven  ? 

102.  Eleven  are  how  many  times  seven  ? 

103.  Twelve  are  how  many  times  seven  ? 

104.  Thirteen  are  how  many  times  seven  ? 

105.  Fourteen  are  how  many  times  seven  ? 

106.  Fifteen  are  how  many  times  seven  ? 

107.  Sixteen  are  how  many  times  seven? 

108.  When  wheat  is  eight  shillings  a  bushel,  what  is 
one  eighth  of  a  bushel  worth  ?  What  are  two  eighths 
of  a  bushel  worth  ?  What  are  three  eighths  of  a  bushel 
worth  ?  What  are  four  eighths  o.f  a  bushel  worth  ? 
Five  eighths  ?     Six  eighths  ?     Seven  eighths  ? 

109.  When  wood  is  eight  dollars  a  cord,  what  part 
of  a  cord  can  you  buy  for  a  dollar  ?  What  part  of  a 
cord  can  you  buy  for  two  dollars  ?  What  part  for  three 
dollars  ?  What  part  for  four  dollars  ?  What  part  for 
five  dollars  ?  What  part  for  six  dollars  ?  What  part 
for  seven  dollars?  How  much  can  you  buy  for  nine 
dollars  ?  How  much  for  ten  dollars  ?  How  much  for 
eleven  dollars  ?  How  much  for  thirteen  dollars  ?  How 
much  for  fifteen  dollars  ?  How  much  for  nineteen  dol- 
lars? 

110.  What  part  of  eight  is  one? 

111.  What  part  of  eight  is  two? 

112.  Three  is  what  part  of  eight? 

113.  Four  is  what  part  of  eight? 

114.  Five  is  what  part  of  eight? 


SEC.iii.]  ARITHMETIC.  43 

115.  What  do  jou  understand  by  one  eighth,  two 
eighths,  etc.,  of  any  number  ? 

IIG.  Seven  is  what  part  of  eight? 

117.  How  many  eighths  make  a  whole  one? 

118.  Ten  are  how  many  times  eight? 

119.  Eleven  are  how  many  times  eight? 

120.  Twelve  are  how  many  times  ei^ht? 

121.  Thirteen  are  how  many  times  eight?      '' 

122.  Fourteen  are  how  many  times  eight? 

123.  When  sugar  is  nine  dollars  a  hundred  weight, 
what  IS  one  ninth  of  a  hundred  weight  worth?  What 
are  two  ninths  of  a  hundred  weight  worth?  Three 
ninths?.  Four  ninths  ?  Five  ninths  ?  Six  ninths^ 
Seventh  ninths?     Eight  ninths? 

^124.  When  rye  is  nine  shillings  a  bushel,  what  part 
ot  a  bushel  can  you  buy  for  one  shilling?  What  part 
for  two  shillings?  What  part  for  three  shillinog?  For 
four  shillings  ?  For  five  shillings  ?  For  six  shillino-s  f 
Seven  shillings?  Eight  shillings?  How  much  l-or 
ten  shillings?  For  thirteen  shillings?  For  fourteen 
shuhngs?^    Sixteen  shillings?     Twenty  shillings? 

12.5.   What  do  you  understand   by  one  ninth,  two 
ninths,  three  ninths,  etc.,  of  any  number? 

126.  Three  is  what  part  of  nine? 

127.  Four  is  what  part  of  nine  ? 

128.  Five  is  what  part  of  nine  ? 

129.  Seven  is  what  part  of  nine? 

130.  How  many  ninths  make  a  whole  one? 

131.  Thirteen  are  how  many  times  nine? 

132.  Fifteen  are  how  many  times  nine? 

133.  Seventeen  are  how  many  times  nine? 

134.  When  hay  is  ten  dollars  a  ton,  what  is  one  tenth 
of  a  ton  worth  ?  What  are  two  tenths  of  a  ton  worth  ? 
What  are  three  tenths  of  a  ton  worth  ?"  Four  tenths? 
Five  tenths?  Six  tenths?  Seven  tenths?  Eight 
tenths?     Nine  tenths? 

135.  When  sugar  is  ten  dollars  a  hundred  weight, 
what  part  of  a  hundred  weight  can  you  buy  for  one 


U  ARITHMETIC.  [parti. 

dollar?  What  part  for  two  dollars?  What  part  for 
three  dollars?  What  part  for  four  dollars?  What 
part  for  five  dollars?  Six  dollars?  Seven  dollars? 
Eight  dollars?  Nine  dollars?  IIow  much  can  you 
buy  for  eleven  dollars?  Thirteen  dollars?  Fifteen 
dollars?     Seventeen  dollars? 

136.  What  do  you  understand  by  one  tenth,  two 
tenths,  three  tenths,  etc.,  of  any  thing?* 

137.  How  many  tenths  make  a  whole  one?* 

C.  Instead  of  writing  the  names  of  numbers,  it  is 
usual  to  express  them  by  particular  characters,  called 
figures. 

One  is  written 1 

Two  is  written. 2 

Three  is  written 3 

Four  is  written 4 

Five  is  written 5 

Six  is  written 6 

Seven  is  written 7 

Eight  is  written 8 

Nine  is  written 9 

Ten  is  written 10 

1.  Eleven  times  one  are  how  many  times  2? 

2.  Twelve  are  how  many  times  2?  3  ?  4? 

3.  Fourteen  are  how  many  times  2?  4?  3? 

4.  If  you  had  fifteen  cents,  how  many  cakes  could 
you  buy  at  4  cents  apiece?  How  many  at  2  cents 
apiece?  How  many  at  3  cents  apiece?  How  many  at 
5  cents  apiece? 

6.  Fifteen  are  how  many  times  4?  2?  3?  5? 

6.  Sixteen  are  how  many  times  5?  3?  6?  2?  7?  4? 

7.  Seventeen  are  how  many  times  6?  2?  7?  3? 
5?  4? 

8.  Eighteen  are  how  many  times  4?  7?  9?  6?  3? 
2?  6?    8? 

*  These  questions  should  frequently  b©  put  to  the  learners. 


SEC.  III.]  ARITHMETIC.  45 

9.  Nineteen  are  how  many  times  3?  7?  4:?  5?  8? 
6?   9?   2?    10? 

10.  Twenty  are  how  many  times  6  ?  2  ?  8  ?  3  ?  9  ? 
4?    10?    6?  7? 

11.  Twenty-one^are  how  many  times  7?  3?  8?  2? 
4?  6?   9?    5?    10? 

12.  Twenty-two  are  how  many  times  3?  8?  5?  4? 
9?   6?   7?    10?    2? 

13.  If  you  had  twenty-seven  dollars,  how  much 
cloth  could  you  buy  at  9  dollars  a  yard  ?  How  much 
at  6  dollars  a  yard?  How  much  at  4  dollars  a  yard? 
How  much  at  3  dollars  a  yard  ?  How  much  at  7  dollars 
a  yard?  How  much  at  8  dollars  a  yard?  How  much 
at  5  dollars  a  yard?     How  much  at  10  dollars  a  yard? 

14.  Twentvscven  are  how  many  times  9?  6?  4? 
3?   7?    8?    5"?    10? 

15.  Twenty-four  are  how  many  times  6?  8?  7? 
6?   2?    10?    3?   4?   9? 

16.  Twenty-nine  are  how  many  times  3?  7?  5? 
9?    6?    8?    4?    10? 

17.  Twenty-three  are  how  many  times  4?  2?  7? 
8?  3?   9?    6?    5?    10? 

18.  Twenty-five  are  how  many  times  3?  7?  2?  6? 
9?   4?    8?    5?    10? 

19.  Thirty  are  how  many  times  10?  2?  3?  7?  9? 
6.?   5?   4?    8? 

20:  Thirty-three  are  how  many  times  6?  8?  7? 
4?    9?    5?    10?    3? 

21.  Twenty-six  are  how  many  times  9?  4?  7?  3? 
8?    5?    6?    10? 

22.  Thirty-five  are  how  many  times  5?  6?  3?  7? 
9?    10?    4?    8? 

23.  Thirty-eight  are  how  many  times  8?  6?  3? 
9?   5?   4?    7?    10? 

24.  Thirty-four  are  how  many  times  7?  3?  9?  10? 
6?    8?   4?   5? 

25.  Thirty-six  are  how  many  times  8?  9?  4?  5? 
3?   6?   7?   10? 


46  ARITHMETIC.  [paet  i. 

26.  Forty  are  how  many  times  8?  10?  6?  4?  3? 
9?  5?  I'i 

27.  For  forty-seven  cents,  how  many  pounds  of 
meat  can  be  bought  at  6  cents  a  pound?  How  many 
pounds  at  8  cents  ?  How  many  at  9  cents  ?  How 
many  at  3  cents  ?  How  many  at  5  cents  ?  How  many 
at  4  cents  ?  How  many  at  7  cents  ?  How  many  at  10 
cents^a  pound? 

28.  Forty-seven  are  how  many  times  6?  8?  9?  3? 
5?   4?    7?    10? 

29.  Forty-three  are  how  many  times  9?  8?  7?  6? 
4?    3?    5?    10? 

30.  Forty-five  are  how  many  times  10?  8?  3?  6? 
4?    7?    5?    9? 

31.  Forty-nine  are  how  many  times  6  ?  10?  6?  9? 
4?    8?    7? 

32.  Fifty-three  are  how  many  times  8?  5?  6?  4? 
7?   9?    10? 

33.  Fifty-seven  are  how  many  times  9  ?  7?  10?  6? 
5?   8?   4? 

34.  Fifty-five  are  how  many  times  0?  4?  8?  10? 
9?    7?    5? 

35.  Forty-eight  are  how  many  times  7?  5?  9?  4? 
6?    8?    10? 

36.  Fifty-four  are  how  many  times  5?  9?  6?  4? 
7?   10?   8? 

37.  Forty-four  are  how  many  times  4?  6?  9?  7? 
5?   8?    10? 

38.  Fifty-eight  are  how  many  times  7  ?  6?  8?  4? 
9?    5?   10? 

39.  Forty  six  are  how  many  times  8?  10?  4?  6? 
9?    7?    5?  ^ 

40.  Fifty  arc  how  many  times  9?  5?  4?  10?  8? 
6?    7? 

41.  Fifty-nine  are  how  many  times  4?  8?  7?  6? 
10?  9?  5? 

42.  Sixty-four  are  how  many  times  7?  5?  8?  10? 
6?  9? 


SEC.    III.]  ARITHMETIC.  47 

43.  Sixty-eight  are  how  many  times  6?  8?  9?  7? 
10?  5? 

44.  Fifty-two  arc  how  many  times  4?  6?  8?  10? 
6?  7?  9? 

45.  Sixty-three  are  how. many  times  5?  4?  6?  10? 
9?  7?  8? 

46.  Sixty-two  are  how  many  times  4?  10?  9?  7? 
8?  5?  6? 

47.  Seventy-three  are  how  many  times  10  ?  7  ?  8  ? 
6?  5?  9? 

48.  Seventy-five  arc  how  many  times  7?  8?  10?  5? 
6?  9? 

49.  If  you  had  sixty-seven  dollars,  how  many  barrels 
of  flour  could  you  buy  at  5  dollars  a  barrel  ?  How 
many  at  seven  dollars  a  barrel?  How  many  at  6  dol- 
lars a  barrel  ?  How  many  at  8  dollars  a  barrel  ?  How 
many  at  10  dollars  a  barrel  ?  How  many  at  9  dollars  a 
barrel  ? 

50.  Sixty-seven  are  how  many  times  5?  7?  6?  8? 
10?  9? 

51.  Seventy-four  are  how  many  times  10?  7?  8?  6? 
5?  9? 

52.  Sixty  are  how  many  times  9?  10?  6?  4?  7? 
6?  8? 

53.  Seventy-two  are  how  many  times  5?  7?  6?  8? 
10?  9? 

54.  Sixty-five  are  how  many  times  5?  10?  8?  6? 
7?  9? 

55.  Sixty-one  are  how  many  times  4?  5?  7?  6?  8? 
10?  9? 

56.  Seventy-nine  are  how  many  times  10?  9?  8? 
7?  6?  5? 

57.  Seventy  are  how  many  times  9?  5?  6?  8? 
7?  10? 

58.  Eighty-two  are  how  many  times  10?  7?  8?  6? 
9? 

59.  Sixty-six  are  how  many  times  9?  5?  6?  7? 
10?  8? 


48  ARITHMETIC.  [part  i. 

60.  Eighty  are  how  many  times  10?  7?  6?  8?  9? 

61.  Sixty-nine  are  how  many  times  9  '/  5  ?  7  ?  10  ? 
8?  6? 

62.  Eiehty-one  are  how  many  times  10?  6?  8?  7? 
9? 

63.  Seventy -six  are  how  many  times  9  ?  5?   10?  6? 
7?  8? 

64.  Eighty-three  are  how  many  times  10  ?   6  ?    7  ? 
9?  8? 

65.  Seventy-one  are  how  many  times  9?  5?  7?  6? 
8?  10? 

66.  Eighty-four  are   how  many  times   10  ?    6  ?    8  ? 
9?  7? 

67.  Seventy-seven  are  how  many  times  9?  7?  5? 
10?  8?  6? 

68.  Eighty-five   are    how  many  times  10  ?    8  ?    7  ? 
6?  9? 

69.  Ninety  are  how  many  times  9?  10?  6?  7?  8? 

70.  Eighty-six  are  how  many  times  10  ?  9  ?  6  ?  7  ? 
8? 

71.  Ninety-four  are  how  many  times   9  ?    10  ?    8  ? 
6?  7? 

72.  Eighty-seven  are  how  many  times  10?  9?    7? 
6?  8? 

73.  Ninety-two  are  how  many  times  9?  10?  6?  7? 
8?  ^ 

74.  Eighty-eight  are  how  many  times  10  ?    9  ?    8  ? 
6?  7? 

75.  Ninety-five  are  how  many  times  9?  10?  6?  8? 
7? 

76.  Eighty-nine  are  how  many  times  10  ?    9  ?    6  ? 
7?  8? 

77.  Ninety-eight  are  how  many  times  10  ?    9  ?    8  ? 
6?  7? 

78.  Ninety-four    are    how  many  times   9  ?   10  ?    6  ? 
8?  7? 

79.  One  hundred  are  how  many  times  6  ?    10  ?    8  ? 
9?  7? 


SEC.  in.]  ARITHMETIC.  49 

80.  Ninety-three  are  how  many  times  10  ?    6  ?  9  ? 
8?  7? 

81.  Ninety-nine  are  how  many  times   7?    10?    8? 
6?  9? 

82.  Ninety-six  are  how  many  times  9?    8?  7?  6? 
10? 

83.  Ninety-seven  are  how  many  times  10  ?  9  ?  6  ? 
7?  8? 

I).  1.  If  an  orange   is  worth  3  apples,  how  many 
oranges  are  fiftccen  apples  worth  ? 

2.  In  8  pints  how  many  quarts? 

3.  In  8  gills  how  many  pints  ? 

4.  If  you  divide  twelve  apples  equally  among  throe 
boys,  how  many  would  you  give  thorn  apiece  ? 

5.  How  many  hours  Would  it  take  you  to  travel  10 
miles,  if  you  travel  three  milt^s  in  an- hour  ? 

6.  How  many  pence  are  there  in  eight  farthings? 

7.  How  many  pence  are  there  in  twelve  farthings  ? 

8.  How  many  pence  are  there  in  seventeen  farthings  ? 

9.  How  many  gallons  are  there  in  ten  quarts  ? 

10.  How  much  broadcloth,  at  six  dollars  a  yard,  can 
you  buy  for  seventeen  dollars  ? 

11.  How  many  pounds  of  raisins,  at  8  cents  a  pound, 
can  you  buy  for  twenty-five  cents  ? 

,  '12.  In  twenty-eight  shillings  how  many  dollars  ?* 

13.  In  twenty-eight  farthings  how  many  pence  ? 

14.  How  many  barrels  of  flour,  at  7  dollars  a  barrel, 
can  you  buy  for  thirty-four  dollars  ? 

15.  How  many  reams  of  paper,  at  five  dollars  a  ream, 
can  you  buy  for  thirty-seven  dollars  ? 

16.  In  thirty-four  gills  how  many  pints? 

17.  In  twenty-seven  quarts  how  many  gallons  ? 

18.  If  an   orange   is  worth    six   apples,  how  many 
oranges  can  you  buy  for  forty  apples  ? 

19.  Thirty-six  shillings  are  how  many  dollars  ? 

*Iu  some  States  8  shillings  is  equal  to  one  dollar. 


50  ARITHMETIC.  [part  i. 

20.  A  man  bought  thirty  apples  at  the  rate  of  3  for 
a  cent,  hew  many  cents  did  they  come  to  ? 

21.  A  laborer  engaged  to  work  8  months  for  ninety- 
six  dollars;  how  much  did  he  receive  for  a  month?  how 
much  a  week,  allowing  4  weeks  to  the  month  ?  how 
many  shillings  a  day,  allowing  6  working  days  to  the 
week  ? 

22.  If  wine  is  worth  twenty  cents  a  pint,  what  is  1 
gill  worth  ? 

23.  If  you  buy  a  bushel*  of  apples  for  forty  cents, 
what  is  the  price  of  a  peck  ? 

24.  If  you  buy  a  bushel  of  pears  for  forty-eight  cents, 
what  will  be  the  price  of  half  a  peck,  at  the  same  rate? 

25.  Four  men  bought  a  horse  for  forty-eight  dollars ; 
what  did  each  man  pay? 

26.  Five  men  bought  a  horse  for  seventy-five  dollars, 
and  sold  him  again  for  forty  dollars ;  what  did  each 
man  lose  by  the  bargain  ? 

27.  A  man  gave  sixty-three  cents  for  a  horse  to  ride 
nine  miles ;  what  was  the  price  of  one  mile's  ride  ? 

28.  A  man  hired  a  horse  to  ride,  and  agreed  to  give 
8  cents  a  mile  ;  he  paid  fifty-six  cents  :  how  many  miles 
did  he  ride  ? 

29.  A  man  had  forty-two  dollars,  which  he  paid  for 
wood  at  7  dollars  a  cord ;  how  many  cords  did  he  buy  ? 

80.  Two  boys  are  forty-eight  rods  apart,  and  both 
running  the  same  way;  but  the  hindmost  boy  gains 
upon  the  other  3  rods  in  a  minute;  in  how  many 
minutes  will  he  overtake  the  foremost  boy? 

31.  There  is  a  vessel  containing  sixty-three  gallons 
of  wine;  it  has  a  pipe  which  discharges  7  gallons  in 
an  hour;  how  many  hours  will  it  take  to  empty  the 
vessel? 

22.  There  is  a  vessel  containing  eighty-seven  gal- 
lons, and  by  a  cock  ten  gallons  will  run  into  it  in  an 
hour;  in  how  many  hours  will  the  vessel  be  filled? 

S3.  If  one  man  can  do  a  piece  of  work  in  thirty 

*1  bushel  is  4  pecks. 


SEC.  IV.]  ARITHMETIC.  51 

days,  in  how  many  days  can  3  men  do  it?  in  how  many 
days  can  5  men  do  it? 

84.  If  you  wish  to  put  sixty-four  pounds  of  butter 
into  8  boxes,  how  many  pounds  would  you  put  into 
each  box? 

35.  If  you  had  seventy-two  pounds  of  butter,  which 
you  wished  to  put  into  boxes  containing  8  pounds 
each,  how  many  boxes  would  it  take? 

36.  If  a  man  can  perform  a  journey  in  thirty-six 
hours,  how  many  days  will  it  take  him  to  do  it  when 
the  days  are  nine  hours  long? 

37.  If  a  man  can  do  a  piece  of  work  in  forty-eight 
hours,  how  many  days  would  it  take  him  to  do  it,  if  he 
works  twelve  hours  in  a  day  ? 


SECTION  IV. 

A.  1.  At  two  cents  a  yard,  what  will  three  yards 
and  one  half  of  a  yard  of  tape  cost? 

2.  3  times  2  and  one  half  of  2  are  how  many? 

3.  At  three  dollars  a  yard,  what  will  4  yards  and  one 
third  of  a  yard  of  cloth  cost? 

4.  4  times  3  and  one  third  of  3  are  how  many? 

5.  At  three  dollars  a  barrel,  what  will  3  barrels  and 
2  thirds  of  a  barrel  of  cider  cost? 

6.  3  times  3  and  two  thirds  of  3  are  how.many? 

7.  If  a  man  earn  4  dollars  in  a  week,  how  many 
dollars  will  he  earn  in  3  weeks  and  1  fourth  of  a  week? 

8.  3  times  4  and  1  fourth  of  4  are  how  many?  _ 

9.  tf  a  yard  of  cloth  cost  4  dollars,  what  will  5 
yards  and  3  fourths  of  a  yard  cost? 

10.  5  times  4  and  3  fourths  of  4  are  how  many? 

11.  If  a  man  spend  five  dollars  in  a  week,  how  many 
dollars  will  he  spend  in  3  weeks  and  1  fifth  of  a^\:eek? 
How  much  in  5  weeks  and  2  fifths  of  a  week? 

12.  3  times  5  and  1  fifth  of  5  are  how  many? 

13.  5  times  5  and  2  fifths  of  5  are  how  many? 


62  ARITHMETIC.  [part  i. 

14.  6  times  5  and  8  fifths  of  5  are  liow  many? 

15.  If  beer  is  worth  six  dollars  a  barrel,  what  would 
4  barrels  and  1  sixth  of  a  barrel  cost?  How  much 
would  7  barrels  and  5  sixths  of  a  barrel  cost? 

IG.  4  times  6  and  1  sixth  of  six  are  how  many? 

17.  7  times  six  and  5  sixths  of  six  are  how 
many  ? 

18.  At  7  dollars  a  barrel,  what  will  o  barrels  and  1 
seventh  of  a  barrel  of  flour  ti'ost?  What  will  5  barrels 
and  2  sevenths  of  a  barrel  cost? 

19.  o  times  7  and  1  seventh  of  7  are  how  many? 

20.  5  times  7  and  2  sevenths  of  7  are  how  many? 

21.  8  times  5  and  4  fifths  of  5  are  how  niany? 

22.  8  times  6  and  3  sixths  of  6  arc  how  ma-ny? 

23.  At  8  dollars  a  yard,  what  will  4  yards  and  1 
eighth  of  a  yard  of  broadcloth  cost? 

24.  4  times  8  and  1  eighth  of  8  are  how  many? 

25.  2  times  7  and  3   sevenths  of  7  are  how  m9,ny? 

26.  8  times  7  and  4  sevenths  of  7  are  how  many? 

27.  9  times  7  and  6  sevenths   of  7  are  how  many? 

28.  3  times  8  and  5  eighths  of  8  are  how  many? 

29.  9  times  8  and  7  eighths  of  8  are  how  many? 

30.  If  a  hundred  weight  of  sugar  cost  9  dollars, 
what  will  2  hundred  weight  and  1  ninth  of  a  hundred 
weight  cost?  What  will  5  hundred  weight  and  2 
ninths  of  a  hundred  weight  cost? 

31.  2  times  9  and  1  ninth  of  9  are  how  many? 

32.  5  times  9  and  2  ninths  of  9  are  how  many? 

33.  6  times  9  and  4  ninths  of  9. are  how  many?    . 

34.  2  times  10  and  3  tenths  of  10  are  how  maqy? 

35.  7  times  9  and  7  ninths  of  9  arc  how  many? 

36.  5  times  10  and  4  tenths  of  10  are  how  many? 

37.  8  times  9  and  5  ninths  of  9  are  how  many? 

38.  4  times  10  and  7  tenths  of  10  are  how  many? 

39.  6  times  10  and  9  tenths  of  10  are  how  many? 

B.  1.  A  man  bought  2  oranges  at  6  cents  apiece; 
how  many  cents  did  they  come  to  ?     He  paid  for  them 


SEC.  IV.]  ARITHMETIC.  63 

with  cherries  at  four  cents  a  pint  j  how  many  pints  did 
it  take? 

2.  2  times  6  are  how  many  times  4? 

3.  A  man  bought  3  yards  of  cloth  at  4  dollars  a 
yard;  how  many  dollars  did  it  come  to?  How  much 
flour  at  6  dollars  a  barrel  would  it  take  to  pay  for  it? 

4.  3  times  4  are  how  many  times  6? 

5.  A  man  boui2;ht  4  peaches  at  3  cents  apiece;  how 
many  cents  did  they  come  to?  He  paid  for  them  with 
pears  at  2  cents  a  piece;  how  many  pears  did  it  take? 

6.  4  times  3  are  how  many  times  2? 

7.  Bought  2  hundred  weight  of  sugar,  at  9  dollars 
a  hundred  weight,  and  paid  for  it  with  wood.'  at  6  dol- 
lars a  cord;  how  many  cords  did  it  take? 

8.  2  times  9  are  how  many  times  G? 

9.  Bought  3  barrels  of  flour  at  8  dollars  a  barrel, 
and  paid  for  it  with  cider  at  4  dollars  a  barrel;  how 
many  barrels  did  it  take? 

10.  3  times  8  are  how  many  times  4? 

11.  12  times  3  are  how  many  times  5? 

12.  6  times  4  are  how  many  times  8? 

13.  3  times  10  are  how  many  times  ,6? 

14.  4  times  9  arcliow  many  times  6? 

15.  How  much  flannel  worth  4  shillings  a  yard 
must  be  given  for  3  yards  of  silk  worth  5  shillings  a 
yard  ? 

16.  3  times  5  are  how  many  times  4  ? 

17.  2  times  7  are  how  many  times  3?  5?  4? 

18.  4  times  5  are  how  many  times  3  ?  6?  7? 

19.  3  times  7  are  how  many  times  4  ?  5?  6?  8?  9? 

20.  Bought  2  kegs  and  2  sevenths  of  a  keg'of 
tobacco  at  7  dollars  a  keg,  and  paid  for  it  with  wood  at 
4  dollars  a  cord;  how  many  cords  did  it  take?  How 
much  butter  at  3  dollars  a  box  would  it  take  to  pay 
for  it? 

21.  2  times  7,  and  2  sevenths  of  7  are  how  many 
times  4?  3?  5?  6?  8? 

22.  Bought  3  bushels  and  3  fifths  of  a  bushel  of 


5i  ARITHMETIC.  [part  i. 

corn  at  5  shillings  a  bushel,  and  paid  for  it  with  wheat 
at  6  shillings  a  bushel;  how  many  bushels  of  wheat 
did  it  take? 

28.  3  times  5  and  3  fifths  of  five  are  how  many 
times  6?   9?  4?   7?   3?   8? 

24.  How  much  sugar  that  is  8  dollars  a  hundred 
weight,  can  be  bought  for  4  cords  and  2  sevenths  of  a 
cord  of  wood,  at  7  dollars  a  cord  ? 

25.  4  times  7  and  2  sevenths  of  7  are  how  many 
times  6?  8?  5?  3?  9?  10'/ 

26.  5  times  5  and  3  fifths  of  5  are  how  many 
times  4?  8?  9?  7?  10?  3?  6? 

27.  6  times  7  and  3  sevenths  of  7  are  how  many 
times  9?  4?  5?  8?  10? 

28.  5  times  8  and  3  eighths  of  8  are  how  many 
times  6?  9?  4?  7?  10? 

29.  7  times  8  and  5  eighths  of  8  are  how  many 
times  9?  6?  10?  4?  5? 

30.  5  times  9  and  4  ninths  of  9  are  how  many 
times  7?  8?  6?  10?  4? 

31.  7  times  9  and  7  ninths  of  9  are  how  many 
times  6?  8?  10?  5?  4? 

32.  6  times  10  and  3  tenths  of  10  are  how  many 
times  7?  5   4?  9?  8? 

33.  8  times  10  and  4  tenths  of  10  are  how  many 
times  6  ?  7  ?  9  ? 

34.  8  times  9  and  3  ninths  of  9  are  how  many 
times  6  ?  10  ?  7  ? 

C.  1.  Bought  4  bushels  of  apples,  at  3  shillings  a 
bushel;  how  many  dollars  did  they  come  to? 

2.  How  many  apples,  at  2  cents  apiece,  must  you 
give  for  2  lemons,  at  4  cents  apiece? 

3.  How  many  pears,  at  3  cents  apiece,  must  you 
give  for  3  oranges,  at  5  cents  apiece  ? 

4.  How  many  barrels  of  cider,  at  3  dollars  a  barrel, 
must  be  given  for  5  boxes  of  butter,  at  4  dollars  a 
box? 


SEC.  IV.]  ARITHMETIC.  55 

5.  A  man  bought  4  yards  of  broadcloth,  at  7  dollars 
a  yard,  and  paid  for  it  with  flour,  at  5  dollars  a  barrel; 
how  many  barrels  did  he  give? 

6.  If  2  apples  cost  4  cents,  what  would  4  apples 
cost? 

7.  If  3  apples  are  worth  6  cents,  how  many  apples 
must  you  give  for  8  pears,  that  are  worth  3  cents 
apiece  ? 

8.  James  had  8  oranges  that  were  worth  5  cents 
apiece,  and  George  had  5  quarts  of  cherries  that  were 
worth  G  cents  a  quart,  which  he  gave  to  James  for  a  part 
of  his  oranges;  how  many  oranges  did  he  buy,  and  how 
many  had  James  left? 

9.  Bought  8  yards  of  cloth,  at  9  shillings  a  yard; 
how  many  dollars  did  it  come  to? 

10.  Bought  5  bushels  and  3  sevenths  of  a  bushel  of 
salt,  at  7  shillings  a  bushel ;  how  many  dollars  did  it 
come  to? 

11.  Bought  9  boxes  and  2  thirds  of  a  box  of  raisins 
for  3  dollars  a  box,  and  paid  for  it  with  cider,  at  4  dol- 
lars a  barrel;  how  many  barrels  did  it  take? 

12.  Bought  8' pounds  and  4  sevenths  of  a  pound  of 
opium,  at  7  dollars  a  pound,  and  paid  for  it  with  cloth, 
at  5  dollars  a  yard;  how  many  yards  did  it  take? 

13.  Bought  (3  chaldrons  and  4  ninths  of  a  chaldron 
of  coal,  at  9  dollars  a  chaldron,  and  paid  for  it  with 
oranges,  at  5  dollars  a  box  ;  how  many  boxes  did  it  take  ? 

14.  Bought  7  cases  and  5  sixths  of  a  case  of  Flo- 
rence oil,  at  6  dollars  a  case,  and  paid  for  it  with  sheet 
lead,  at  7  dollars  a  hundred  weight;  how  many  hundred 
weight  did  it  take? 

15.  liow  inany  dozen  of  eggs,  at  8  cents  a  dozen, 
must'  you  give  for  7  pounds  of  sugar,  at  10  ocnts  a 
pound  ? 

16.  How  much  barley,  at  3  shillings  a  bushel,  must 
be  niven  for  8  bushels  of  wheat,  at  7  shillings  a  bushel  ? 

17.  How  much  cloth,  at  4  shillings  a  yard,  must  be 
given  for  a  firkin  of  butter  worth  8  dollars  ? 


56  ARITHMETIC.  [part  i. 

18.  IIoTT  much  clotli,  at  5  shillings  a  yard,  can  be 
bought  for  2  reams  of  paper,  at  5  dollars  a  ream? 

19.  IIow  much  -wha'at,  at  7  shillings  a  bushel,  can 
be  bought  for  2  barrels  of  cider,  at  4  dollars  and  a 
half  a  barrel? 

20.  flow  long  would  it  take  a  man  to  lay  up  10 
dollars,  if  he  saves  4  shillings  a  day? 

21.  If  a  man  earns  8  shillings  a  day,  how  many  dol- 
lars would  he  earn  in  10  days  ? 

22.  A  man  bought  twenty  pears  at  the  rate  of  2 
for  3  cents;  how  much  did  tjiey  come  to? 

23.  How  many  eggs,  at  the  rate  of  3  for  5  cents, 
can  you  buy  for  thirty  cents  ? 

24.  A  man  hired  a  laborer  and  agreed  to  give  him 
5  dollars  for  every  3  days'  work ;  how  much  did  he 
give  him  a  week,  there  being  G  working  days  in  a 
week  ?  How  much  was  it  a  month,  allowing  4  weeks 
to  a  month  ? 

25.  If  a  man  receives  5  dollars  for  3  days'  work, 
how  many  shillingi?  is  that  a  day  ? 

26.  5  men  bought  a  horse  for  sixty-three  dollars, 
and  paid  two  dollars  a  week  for  keening  him;  at  the 
end  of  8  weeks  they  sold  him  for  fifty-four  dollars; 
how  much  did  each  man  lose  by  the  bargain  ? 

SECTION  y. 

A.  1.  James  had  4  apples,  and  John  had  half  as 
many;  how  many  had  John  ? 

2.  If  an  orange  cost  6  'cents,  and  an  apple  half  as 
much  ;   how  much  docs  the  apple  cost  ? 

3.  If  you  divide  8  apples  equally  betwijen  two  boys, 
^<'hat  part  of  them  must  each  have  ?       • 

Ans.   One  half  of  them. 

4.  What  is  1  half  of  8  ? 

5.  If  you  divide  8  apples  equally  among  4  boys, 
what  part  of  them  must  each  have  ? 

Ans.  One  fourth  of  them. 


SEC.  v.]  ARITHMETIC.  57 

6.  What  is  1  fourth  of  8  ? 

7.  If  you  divide  6  oranges  equally  among  3  boys, 
what  part  of  thciii  must  1  boy  have  ? 

8.  What  is  1  third  of  6? 

9.  If  4  yards  of  cloth  cost  8  dollars,  vv^hat  part  of 
8  dollars  would  1  yard  cost?  What  part  of  8  dollars 
would  2  yards  cost?  Vv'hat  part  of  8  dollars  would  o 
yards  cost  ? 

10.  What  is  1  fourth  of  8  ?  What  is  2  fourths  of 
8?     What  is  3  fourths  of  8? 

11.  If  G  yards  of  cloth  will  make  8  coats,  what  part 
of  G  yards  will  make  1  coat  ?  What  part  of  G  3^ards 
will  make  two  coats  ? 

12.  What  is  one  third  of  6?  What  is  two  thirds 
of  G? 

13.  If  3  barrels  of  cider  cost  9  dollars,  w^hat  part 
of  9  dollars  will  1  barrel  cost  ?  What  part  of  9  dol- 
lars will  2  barrels  cost  ? 

14.  What  is  1  third  of  9  ?     What  is  2  thirds  of  9  ? 

15.  If  2  yards  of  cloth  cost  10  dollars,  what  part  of 
10  dollars  will  1  yard  cost  ?  What  part  of  10  dollars 
will  3  yards  cost  ? 

IG.  What  is  1  half  of  10?  What  is  3  halves  of 
10? 

17.  If  2  barrels  of  flour  cost  twelve  dollars,  what 
part  of  twelve  dollars  will  1  barrel  cost?  What  part 
of  twelve  dollars  will  3  barrels  cost  ?  What  part  of 
twelve  dollars  will  5  barrels  cost  ? 

18.  What  is  1  half  of  twejve  ?  What  is  3  halves 
of  twelve  ?     What  is  5  halves  of  twelve  ? 

19.  If  4  barrels  of  cider  cost  twelve  dollars,  what 
part  of  twelve  dollars  will  1  barrel  cost?  What  part 
of  twelve  dulhirs  will  3'  barrels  cost  ?  What  part  of 
twelve  dollars  will  5  barrels  cost?  Wliat  part  of 
twelve  dollars  will  7  barrels  cost  ? 

20.  What  is  1  fourth  of  twelve?  What  is  2  fourths 
of  twelve  ?  3  fourths  ?  5  fourths  ?  7  fourths  ? 

21.  If  three  oranges  cost  twelve  cents,  what  part  of 


58  ARITHMETIC.  [part  i. 

twelve  cents  will  1  orange  cost  ?  What  part  of  twelve 
cents  will  2  oranges  cost  ?  What  part  of  twelve  cents 
will .4  oranges  cost?  What  part  of  twelve  cents  will 
5  oranges  cost  ?    7  oranges  ?    1.0  oranges  ? 

22.  What  is  one  third V  twelve?  2  thirds?  4  thirds? 

5  thirds?  7  thirds?  10  thirds? 

23.  If  five  bushels  of  wheat  cost  10  dollars,  what 
part  of  10  dollars  will  1  bushel  cost  ?  What  part  of 
10  dollars  will  2  bushels  cost  ?  3  bushels  ?  4  bushels  ? 

6  bushels  ?  7  bushels  ? 

24.  What  is  1  fifth  of  ten?  2  fifths?  3  fifths?  4 
fifths  ?  6  fifths  ?   7  fifths  ? 

25.  What  is  1  half  of  fourteen  ?  1  seventh  ?  2  sev- 
enths ?  3  sevenths  ?  5  sevenths  ? 

26.  What  is  1  third  of  fifteen  ?  2  thirds  ?  1  fifth  ? 

2  fifths  ?  3  fifths  ?  4  fifths  ? 

27.  What  is  1  half  of  sixteen?  3  fourths?  1  eighth? 

3  eii^hths?  5  eighths?  7  eighths?  1  sixteenth? 

28.  What  is  1  half  of  eighteen  ?.  2  thirds  ?  1  sixth? 
5  sixths  ?  1  ninth  ?  2  ninths?  4  ninths?  5  ninths?  7 
ninths  ?  8  ninths  ?   1  eighteenth  ?  5  eighteenths  ? 

29.  What  is  1  half  of  twenty  ?  1  fourth  ?  3  fourths  ? 
1  fifth  ?  3  fifths  ?  2  fifths  ?  4  fifths  ?  1  twentieth  ?  3 
twentieths  ?  7  twentieths  ? 

30.  What  is  1  third  of  twenty-one  ?  2  thirds  ?  1 
seventh  ?  5  sevenths  ?  3  sevenths  ?  6  sevenths  ?  4 
sevenths  ? 

31.  What  is  1  half  of  twenty-two?  1  eleventh?  3 
elevenths  ?  5  elevenths  ?   7  elevenths  ?  10  elevenths  ? 

32.  What  is  1  half  of  twenty-four?  1  third?  2 
thirds  ?  1  fourth  ?  3  fourths  ?  1  sixth  ?  5  sixths  ?  1 
eighth?  3  eighths?  7  eighths?  1  twelfth?  5  twelfths? 

7  twelfths  ?  " 

33.  What  is  1  half  of  twenty-six? 

34.  What  are  2  thirds  of  twenty-seven?  2  ninths? 

4  ninths?  5  ninths?  8  ninths?  7  ninths? 

35.  What  is  1  half  of  twenty-eight?  3  fourths?  2 
sevenths  ?  5  sevenths  ?  3  sevenths  ? 


SEC.  v.]  ARITHMETIC.  59 

36.  What  is  1  sixth  of  tliirty?  3  fifths?  7  tenths? 
2  thirds?  1  half? 

37.  What  is  1  half  of  thirty-two  ?  3  fourths  ?  5 
eighths  ?  1  sixteenth  ? 

38.  What  is  1  half  of  thirty-four? 

39.  Whatis  5  sixths  of  thirty-six?  2  thirds?  1  half  ? 
7  twelfths  ?  4  ninths  ?  3  fourths  ? 

40.  Whatis  1  half  of  thirty-eight? 

41.  What  is  1  third  of  thirty-nine  ? 

42.  What  is  3  fourths  of  forty?  7  eighths?  9  tenths? 
1  half? 

43.  AYhat  is  3  sevenths  of  forty-two?  5  sixths? 

B.  1.  A  boy  having  twelve  apples,  kept  1  fourth  of 
them  himself,  and  divided  the  other  3  fourths  of  them 
equally  among  4  of  his  companions;  how  many  did  he 
give  them  apiece  ? 

2.  3  fourths  of  twelve  are  how  many  times  4  ? 

3.  A  man  having  fourteen  bushels  of  grain,  divided 
5  sevenths  of  it  equally  among  3  men ;  how  much  did 
he  give  them  apiece  ? 

4.  5  sevenths  of  fourteen  are  how  many  times  3  ? 

5.  A  man  having  fifteen  shillings,  gave  away  4  fifths 
of  it,  how  many  dollars  did  he  give  away? 

6.  4  fifths  of  nineteen  are  how  many  times  6? 

7.  A  man  having  twenty-one  cents,  paid  away  6  sev- 
enths of  them  for  oranges  at  5  cents  apiece;  how  many 
oranges  did  he  buy  ? 

8.  6  sevenths  of  twenty-one  are  how  many  times  5? 

9.  A  man  bought  a  piece  of  cloth  for  twenty-four 
shillings,  and  sold  it  again  for  5  thirds  of  what  he  gave 
for  it;  how  many  dollars  did  he  sell  it  for? 

10.  5  thirds  of  twenty-four  are  how  many  times  6? 

11.  7  fifths  of  twenty  are  how  many  times  6  ? 

12.  5  sevenths  of  thirty-five  are  how  many  times  8? 

13.  7  sixths  of  thirty-six  are  how  many  times  5  ? 

14.  9  sevenths  of  forty-two  are  how  many  times  8  ? 

15.  7  ninths  of  forty -five  are  how  many  times  6  ? 


GO  ARITHMETIC.  [i'art  i. 

16.  5  sixths  of  forty-eight  are  how  many  times  7  ? 

17.  6  fifths  of  fifty  are  how  many  times  9  ? 

18.  8  ninths  of  fifty-four  are  how  many  times  5  ? 

19.  9  sevenths  of  fifty-six  are  how  many  times   10  ? 

20.  7  sixths  of  sixty  are  how  many  times  8  ? 

21.  9  sevenths  of  sixty-three  are  how  many  times  7  ? 

22.  10  eighths  of  sixty-four  are  how  many  times  9  ? 

23.  6  eighths  of  seventy-two  are  how  many  times  7? 

24.  4  sevenths  af  eighty-four  arc  how  many  times  9? 

25.  7  tenths  of  ninety  are  how  many  times  8  ? 

G.  1.  Charles  had  6  apples,  and  gave  1  third  of  them 
to  John  ;  how  many  did  he  give  him  ? 

2.  Albert  had  9  cents,  and  spent  2  thirds  of  them; 
how  many  had  he  left  ? 

8.  James  had  10  pears,  and  gave  1  half  of  them  to 
one  of  his  companions,  and  2  fifths  of  them  to  another ; 
how  many  did  he  give  away? 

4.  If  3  yards  of  cloth  cost  6  dollars,  what  is  that  a 
yard  ? 

5.  If  4  yards  of  cloth  cost  twelve  dollars,  what  will 
2  yards  cost? 

6.  If  9  apples  cost  eighteen  cents,  what  will  3  apples 
cost  ? 

7.  If  3  oranges  cost  18  cents,  what  will  2  cost  ? 

8.  James  had  twenty-five  cents,  and  he  gave  4  fifths 
of  them  for  10  apples;  how  much  did  he  give  for  all 
the  apples  ?  how  much  apiece  ? 

9.  A  man  had  thirty  dollars,  and  gave  5  sixths  of 
them  for  8  yards  of  cloth ;  how  much  did  he  give  a 
yard  ? 

10.  A  man  had  forty  yards  of  cloth,  and  sold  3  fifths 
of  it  for  twenty-four  dollars ;  what  was  that  a 
yard  ? 

11.  A  man  had  forty-two  barrels  of  flour,  and  sold  2 
sevenths  of  it  for  6  dollars  a  barrel ;  how  much  did  it 
come  to. 

12.  A  boy  had  sixty-three  nuts,  and  divided  4  sev- 


SEC.  v.]  ARITHMETIC.  61 

entlis  of  them  equally  among  six  of  his  companions  ', 
how  many  did  he  give  them  apiece  ? 

13.  If  4  yards  of  cloth  cost  twelve  dollars,  what  will 
three  yards  cost? 

14.  If  five  oranges  cost  twenty-five  cents,  what  will 
3  cost  ? 

15.  If  3  oranges  cost  fifteen  cents,  what  will  7  cost? 

16.  If  three  barrels  of  cider  cost  twelve  dollars,  what 
will  10  barrels  cost  ? 

17.  If  7  pounds  of  flour  cost  thirty-five  cents,  what 
will  nine  pounds  cost  ? 

18.  If  5  firkins  of  butter  cost  forty  dollars,  what 
will  three  firkins  cost? 

19.  If  2  men  can  do  a  piece  of  work  in  six  days, 
how  long  would  it  take  4  men  to  do  the  same  w^ork  ? 

20.  If  6  men  can  do  a  piece  of  work  in  twelve  days, 
in  how  many  days  will  3  men  do  the  same  work  ? 

21.  If  3  men  can  do  a  piece  of  work  in  twelve  days, 
in  how  many  days  will  4  men  do  the  same  work  ? 

22.  If  2  cocks  of  a  aertain  size  will  empty  a  cistern 
in  6  hours,  in  how  long  a  time  will  3  cocks  of  the  same 
size  enjpty  it  ? 

23.  Three  men,  setting  out  on  a  journey,  purchased 
5  loaves  of  bread  apiece,  but  before  they  had  eaten  any 
of  it,  two  other  men  joined  them,  and  they  agreed  to 
share  the  bread  equally  among  the  whole ;  how  many 
loaves  did  they  have  apiece? 

24.  If  4  barrels  of  flour  cost  twenty-four  dollars, 
what  would  7  barrels  cost?  IIow  much  cider  at  3  dol- 
lars a  barrel  would  7  barrels  of  flour  buy  ? 

25.  A  man  bought  a  quantity  of  flour  for  fifty-four 
dollars ;  and  another  man  gave  him  9  yards  of 
cloth  for  5  sixths  of  it;  what  was  the  cloth  worth  a 
yard? 

26.  If  9  yards  of  cloth  cost  fifty-four  dollars,  how 
many  boxes  of  butter,  at  4  dollars  a  box,  would  5  yards 
of  the  same  cloth  buy  ? 

27.  Bought  8  firkins  of  butter  for  seventy-two  doU 


62  ARITHMETIC.  [paet  i 

lars,  and  gave  6  of  them  for  7  yards  of  cloth ;  what 
was  a  yard  of  the  cloth  worth  '/ 

28.  A  man  bought  6  barrels  of  flour  for  sixty  dol- 
lars, and  gave  4  barrels  of  it  for  cider  at  5  five  dollars 
a  barrel ;  how  many  barrels  did  he  buy? 

Note. — The  manner  of  writing  numbers  with  figures 
has  been  explained  as  far  as  ten.  The  numbers  from 
ten  to  one  hundred  are  written  as  follows  : 

Ten  is  written 10 

Eleven 11 

Twelve 12 

Thirteen 13 

Fourteen 14 

Fifteen 15 

Sixteen 16 

Seventeen 17 

Eighteen 18  ^ 

Nineteen 19 

Twenty 20 

Twenty-one '. —  21 

Twenty-two : 22 

Twenty-three 23 

Twenty-four 24' 

Twenty-five 25 

T  wen  ty-six ." 26 

Twenty-seven 27 

Twenty-eight 28 

Twenty-nine 29 

Thirty 30 

Thirty-one,  etc 31 

Forty 40 

Fifty 50 

Sixty 60 

Seventy 70 

Eighty 80 

Ninety 90 

One  hundred 100 


SEC.  VI.]  ARITHMETIC.  63 

SECTION  VI. 

A.  1.  Bought  1  half  of  a  yard  of  cloth  for  I  shil- 
ling, what  would  be  the  price  of  a  yard  at  the  same 
rate  ? 

2.  If  1  half  pint  of  cherries  cost  2  cents,  what  will 
a  pint  cost  ? 

3.  If  1  fourth  of  a  barrel  of  flour  cost  2  dollars,  what 
would  a  barrel  cost  ? 

4.  2  is  1  half  of  what  number  ?  1  fourth  of  what 
number  ? 

5.  If  1  third  of  a  yard  of  cloth  cost  2  dollars,  what 
is  that  a  yard  ? 

6.  2  is  1  third  of  what  number  ? 

7.  If  1  tbird  of  a  yard  of  cloth  cost  3  dollars,  what 
is  that  a  yard  ? 

8.  3  is  1  third  of  what  number  ? 

9.  If  1  fourth  of  a  firkin  of  butter  cost  3  dollars, 
what  is  that  a  firkin  ? 

10.  3  is  1  fourth  of  what  number? 

11.  A  man  bought  1  third  of  a  bushel  of  wheat  for 
4  shillings,  what  would  a  bushel  cost  at  that  rate? 

12.  4  is  1  third  of  what  number? 

13.  If  a  man  can  ride  2  miles  in  1  fifth  of  an  hour, 
how  far  can  he  ride  in  an  hour  ? 

14.  2  is  1  fifth  of  what  number? 

15.  A  man,  being  asked  the  age  of  his  eldest  son, 
answered  that  his  youngest  son,  who  was  3  years  old, 
was  just  I  fifth  of  the  age  of  his  eldest  son;  how  old 
was  the  eldest  son  ? 

16.  3  is  1  fifth  of  what  number?  « 

17.  A  inan  bought  I  sixth  part  of  a  hundred  weight 
of  sugar  for  2  dollars,  what  would  a  hundred  weight 
cost  at  the  same  rate? 

18.2  is  1  sixth  of  what  number  ? 

19.  Bought  1  fifth  of  a  pound  of  starch  for  5  cents, 
what  was  that  a  pound  ? 

20.  5  is  1  fifth  of  what  number? 


64  AllITHMETIC.  [part  i. 

21.  Bought  1  fourth  of  a  pound  of  aquafortis  for  6 
cents,  what  was  that  a  pound  ? 

22.  6  is  1  fourth  of  what  number? 

23.  Bouiiht  1  seventh  of  a  hundred  weight  of  cocoa 
for  4  dollars,  what  would  1  hundred  weight  cost  at  the 
same  rate  ? 

24.  4  is  1  seventh  of  what  number? 

25.  7  is  1  fifth  of  what  number  ? 

26.  5  is  1  third  of  what  number? 

27.  4  is  1  eio-hth  of  what  number? 

28.  6  is  one  sixth  of  what  number? 

29.  8  is  1  third  of  what  number?  • 
80.  9  is  1  fourth  of  what  number  ? 

31.  7  is  1  sixth  of  what  number? 

32.  8  is  1  seventh  of  what  number? 

33.  9  is  1  eighth  of  what  number? 

34.  8  is  1  tenth  of  what  number? 

35.  7  is  1  ninth  of  what  number? 

36.  6  is  1  fifth  of  what  number? 

37.  10  is  1  seventh  of  what  number? 

B.  1.  A  man  bought  some  linen  and  some  cotton 
cloth ;  for  the  linen  he  gave  4  shillings  a  yard,  which 
was  twice  as  much  as  he  gave  for  the  cofton;  what  did 
he  give  a  yard  for  the  corton  ? 

2.  4  is  2  times  what  number? 

3.  If  2  thirds  of  a  yard  of  cloth  cost  6  dollars,  what 
would  1  third  cost? 

4.  6  is  2  times  what  number? 

5.  If  3  fourths  of  a  barrel  of  flour  cost  6  dollars, 
what  wfll  1  fourth  of  a  barrel  cost  ? 

6.  6  is  3  times  what  number? 

7.  If  2  fifths  of  a  pound  of  chocolate  cost  8  cents, 
what  would  1  fifth  of  a  pound  cost? 

8.  8  is  2  times  what  number? 

9.  If  3  fifths  of  a  pound  of  candles  cost  9  cents, 
what  will  1  fifth  of  a  pound  cost? 

10.  9  is  3  times  what  number? 


SEC.  VI.]  ARITHMETIC.  65 

11.  If  2  sevenths  of  a  pound  of  spermaceti  candles 
cost  10  cents,  what  will  1  seventh  of  a  pound  cost? 

12.  10  is  2  times  what  number? 

13.  If  5  ei<jfhths  of  a  pound  of  cotton  cost  10  cents, 
what  will  1  eighth  cost? 

14.  10  is  5  times  what  number? 

15.  If  2  thirds  of  a  yard  of  cloth  cost  4  dollars, 
what  will  1  third  cost?  If  1  third  of  a  yard  Cost  2 
dollars,  what  will  a  yard  cost?* 

16.  If  4  is  2  thirds  of  some  number,  what  is  1  third 
of  the  same  number?  2  is  1  third  of  what  number?* 
Then  4  is  2  thirds  of  what? 

17.  If  2  thirds  of  a  barrel  of  flour  cost  6  dollars, 
what  will  1  third  of  a  barrel  cost?  If  1  third  of  a 
barrel  cost  3  dollars,  what  will  a  barrel  cost?* 

18.  If  6  is  2  thirds  of  some  number,  what  is  1  third 
of  the  same  number?  3  is  1  third  of  what  number? 
Then  6  is  2  thirds  of  what? 

19.  If  3  fourths  of  a  bushel  of  wheat  cost  6  shil- 
lings, what  will  1  fourth  of  a  bushel  cost?  If  1 
fourth  of  a  bushel  cost  2  shillings,  what  will  a  bushel 
cost? 

20.  If  6  is  3  fourths  of  some  number,  what  is  1 
fourth  of  the  same  number?  2  is  1  fourth  of  what 
number?     Then  6  is  3  fourths  of  -what? 

21.  If  2  fifths  of  a  gallon  of  wine  cost  4  shillings, 
what  will  1  fifth  of  a  gallon  cost?  If  1  fifth  of  a  gal- 
lon cost  2  shillings,  what  will  a  gallon  cost? 

22.  If  4  is  2  fifths  of  some  number,  what  is  1  fifth 
of  the  same  number?  2  is  1  fifth  of  what  number? 
Then  4  is  2  fifths  of  what?       ' 

23.  If  3  sevenths  of  a  pound  of  tobacco  cost  6 
cents,  what  will  1  seventh  of  a  pound  cost?  If  1 
seventh  of  a  pound  cost  2  cents,  what  will  a  pound 
cost  ? 

24.  If  6  is  3  sevenths  of  some  number,  what  is  1 


*  See  this  section,  article  A. 


66  ARITHMETIC.  [part  i. 

seventh  of  the  same  number?     2  is  1  seventh  of  what 
number?     Then  6  is  3  sevenths  of  what? 

25.  If  2  sevenths  of  a  barrel  of  fish  cost  4  dollars, 
what  will  1  seventh  of  a  barrel  cost?  What  will  a 
barrel  cost? 

26.  4  is  2  sevenths  of  what  number  ? 

27.  If  3  eighths  of  a  pound  of  chocolate  cost  6 
cents,'' what  will  1  eighth  of  a  pound  cost?  What  will 
a  pound  cost? 

28.  6  is  3  eights  of  what  number? 

29.  If  8  cents  will  buy  2  fifths  of  a  pound  of  aqua- 
fortis, how  many  cents  will  buy  a  pound  ? 

'  30.  8  is  2  fifths  of  what  number? 

31.  A  .man  bought  3  fourths,  of  a  hundred  weight 
of  yellow  ochre  for  9  dollars;  what  was  that  a  hundred 
weight? 

32.  9  is  3  fourths  of  what  number? 

33.  8  is  4  ninths  of  what  number? 

34.  9  is  3  tenths  of  what  number? 

35.  10  is  5  sevenths  of  what  number? 

36.  12  is  3  fifths  of  what  number? 

37.  12  is  4  ninths  of  what  number? 

38.  10  is  2  sevenths  of  what  number? 

39.  14  is  7  fifths  of  what  number? 

40.  15  is  3  elevenths  of  what  number? 

41.  16  is  2  fifths  of  what  number? 

42.  18  is  6  tenths  of  what  number? 

43.  20  is  5  ninths  of  what  number? 

44.  21  is  3  ninths  of  what  number? 

45.  25  is  8  ninths  of  what  number? 

0.  1.  If  5  eighths  of  a  cask  of  claret  wine  cost  15 
dotlars,  what  is  that  a  cask  ?  How  much  cider  at  6 
dollars  a  barrel  would  it  take  to  pay  for  a  cask  of  the  wine? 

2.  15  is  5  eighths  of  how  many  times  4? 

3.  If  2  thirds  of  a  pound  of  coifee  cost  18  cents, 
how  much  would  a  pound  cost?  How  many  ox'anges 
at  5  cents  apiece,  might  be  bought  for  a  pound? 


SEC.    VI.]  ARITHMETIC.  G7 

4.  18  is  2  thirds  of  how  many  times  5? 

5.  A  man  bought  4  sevenths  of  a  hundred  wcii^ht 
of  sugar  for  20  shillings,  how  many  dollars  would  a 
hundred  weight  come  to  at  the  same  rate? 

6.  20  is  4  sevenths  of  how  many  times  6? 

7.  A  man  sold  a  cow  for  21  dollars,  which  was  only 
7  tenths  of  what  she  cost  him  ;  how  much  did  she  cost 
him  ?  When  he  bought  her,  he  paid  for  her  with  cloth 
at  8  dollars  a  yard ;  how  many  yards  of  cloth  did  he 
give  ? 

8.  21  is  seven  tenths  of  how  many  times  8? 

9.  A  mau  being  asked  the  age  of  his  youngest  son, 
answered,  that  the  age  of  his  eldest  son  was  24  years, 
which  was  three  fifths  of  his  own  age;  and  that  his  own 
age  was  10  times  as  much  as  that  of  his  youngest  son; 
what  was  his  age  y  and  what  was  the  age  of  his  young- 
est son  ? 

10.  24  is  3  fifths  of  how  many  times  10? 

11.  27  is  3  fifths  of  how  many  times  7? 

12.  28  is  7  tenths  of  how  many  times  9  ?  ' 

13.  30  is  5  eighths  of  how  many  times  7? 

14.  32  is  4  sevenths  of  how  many  times  6  ? 

15.  36  is  9  eighths  of  how  many  times  5  ? 

16.  40  is  8  ninths  of  how  many  times  8  ? 

17.  42  is  6  fifths  of  how  many  times  4? 

.    18.  45  is  9  eighths  of  how  many  times  6  ? 

19.  48  is  8  ninths  of  how  many  times  7  ? 

20.  *  50  is  5  sevenths  of  how  many  times  8  ? 

21.  54  is  9  sixths  of  how  many  times  7  ? 

22.  56  is  7  ninths  of  how  many  times  10? 

23.  60  is  10  sevenths  of  how  many  times  4  ? 

24.  63  is  9  eighths  of  how  many  times  5  ? 

25.  64  is  8  ninths  of  how  many  times  7  ? 
26.-  70  is  10  sevenths  of  how  many  times  8  ? 

27.  72  is  9  fifths  of  how  many  times  6  ? 

28.  80  is  10  thirds  of  how  many  times  4? 

29.  80  is  8  fifths  of  how  many  times  6  ? 


68  ARITHMETIC.  [parti. 

D.  1.  A  boy  gave  away  4  cents,  which  was  1  third 
of  all  he  had  ;  how  many  had  he  at  J5rst  ? 

2.  A  boy  gave  5  apples  to  one  of  his  companions, 
which  was  1  fourth  of  whatjie  had;  how  many  had  he? 

3.  A  man  paid  away  4  dollars,  which  was  2  thirds  of 
all  the  money  he  had;  how  much  had  he? 

4.  A  man  sold  a  watch  for  18  dollars,  which  was  3 
fourths  of  what  it  cost  him  ;  how  much  did  is  cost  ? 

5.  A  man  sold  a  cow  for  15  dollars,  which  was  3  fifths 
of  what  the  cow  cost ;  how  much  did  he  lose  by  his 
bargain  ? 

6.  A  man  bought  12  yards  of  cloth,  and  sold  it  for 
54  dollars,  which  was  9  eighths  of  what  it  cost  him; 
what  did  it  cost  him  a  yard  ?  and  how  much  did  he  gain 
by  his  bargain  ? 

7.  There  is  a  pole  standing  in  the  water,  so  that  10 
feet  of  it  is  above  the  water,  which  is  2  thirds  of  the 
whole  length  of  the  pole ;  how  long  is  the  pole  ? 

8.  There  is  a  pole  2  thirds  under  water,  and  4  feet 
out;  how  long  is  the  pole? 

9  There  is  a  pole  2  fifths  under  water,  and  6  feet 
out  of  the  water ;  how  long  is  the  pole  ? 

10.  There  is  an  orchard,  in  which  3  sevenths  of  the 
trees  bear  cherries,  and  2  sevenths  bear  pe'aches,  and 
10  trees  bear  plums ;  how  many  trees  are  there  in  the 
orchard  ?  and  how  many  of  each  sort  ? 

11.  There  is  a  school,  in  which  2  ninths  of  the  boys 
learn  arithmetic,  3  ninths  learn  grammar,  1  ninth  learn 
geography,  1  ninth  learn  geometry,  and  12  learn  to 
write ;  how  many  are  there  in  the  school  ?  and  how 
many  attending  to  each  study  ? 

12.  A  man  sold  a  watch  for  63  dollars,  which  was 
7  fifths  of  what  it  cost  him;  how  much  did  he  gain  by 
the  bargain  ? 


SEC.  VI.]  ARITHMETIC.  69 

MISCELLANEOUS   EXAMPLES. 

1.  If  1  yard  of  cloth  cost  4  dollars,  what  will  5  yards 
cost? 

2.  A  man  bought  3  pounds  of  raisins,  at  7  cents  a 
pound,  and  16  oranges,  at  4  cents  apiece,  and  1  pound 
of  candles  for  16  cents;  what  did  they  all  come  to? 

'  3.  A  boy  had  37  apples ;  he  kept  5  himself,  and 
divided  the  rest  equally  among  four  companions  ;  how 
many  did  he  give  them  apiece  ? 

4.  Two  men  are  40  miles  apart,  and  both  travelling 
the  same  way ;  the  hindermost  man  gains  upon  the  other 
5  miles  each  day  ;  in  how  many  days  will  he  overtake 
him  ? 

5.  Two  men  are  travelling  the  same  way,  one  travels 
at  the  rate  of  38  miles  a  day,  the  other,  who  is  behind 
the  former,  travels  44  miles ;  how  much  does  he  gain 
of  the  first  dach  day?  and  in  how  many  days  would  he 
gain  60  miles  ? 

6.  A  fox  is  80  rods  before  a  greyhound,  and  is  run- 
ning at  the  rate  of  27  rods  in  a  minute;  the  greyhound 
is  following  at  the  rate  of  31  rods  in  a  minute;  in 
how  many  minutes  will  the  greyhound  overtake  the  fox? 

7.  If  2  yards  of  cloth  cost  6  dollars,  what  would  4 
yards  cost  ?  what  would  12  yards  cost  ? 

8.  If  8  sheep  cost  24  dollars,  what  would  B  cost  ? 

9.  If  4  tons  of  hay  will  keep  3  horses  through  the 
winter,  how  many  tons  will  keep  30  horses  the  same 
time  ? 

10.  If  a  man  spends  8  shillings  a  day,  how  many  dol- 
lars will  he  spend  in  a  week  ? 

11.  Bought  10  pieces  of  cloth,  each  containing  5 
yards,  for  100  dollars;  what  was  it  a  piece?  and  how 
much  a  yard  ? 

12.  If  the  wages  of  12  weeks  come  to  60  dollars, 
what  is  that  a  month  ?  and  how  much  for  5  weeks  ? 

13.  If  7  horses  eat  14  bushels  of  oats  in  1  week,  how 
many  bushels  would  15  horses  eat  in  the  same  time? 


70  ARITHMETIC.  [part  i. 

14.  If  3  horses  eat  8  bushels  of  oats  in  2  weeks, 
how  long  would  it  take  them  to  eat  40  bushels  ? 

15.  If  1  horse  eat  1  bushel  of  oats  in  3  days,  in  how 
many  days  would  4  horses  eat  36  bushels  ? 

16.  If  2  men  spend  12  dollars  in  1  week,  how  many 
dollars  would  3  men,  at  the  same  rate,  spend  in  5 
weeks  ? 

17.  If  a  staff  3  feet  long  cast  a  shadow  of  2  feet  at 
12  o'clock,  what  is  the  length  of  a  pole  that  casts  a 
shadow  eighteen  feet  at  the  same  time  of  day? 

18.  If  47  gallons  of  water,  in  1  hour,  run  into  a 
cistern  containing  108  gallons,  and  by  a  pipe  38  gallons 
run  out  in  an  hour,  how  much  remains  in  the  cistern 
in  an  hour?  and  in  how  many  hours  will  the  cistern  be 
filled  ? 

19.  If  4  men  can  do  a  piece  of  work  in  8  days,  how 
many  men  would  it  take  to  do  the  same  work  in  4  days? 

20.  If  6  men  can  do  a  piece  of  work 'in  .9  days,  in 
how  many  days  would  2  men  do  it  ? 

21.  If  8  men  can  do  a  piece  of  work  in  5  days,  in 
how  many  days  would  they  do  a  piece  of  work  4  times 
as  large  ? 

22.  If  7  men  can  do  a  piece  of  work  in  3  days,  how 
many  men  would  it  take  to  do  a  piece  of  work  5  times 
as  large  in  the  same  time  ? 

23.  If  8  men  can  do  a  piece  of  work  in  4  days,  in 
how  many  days  would  2  men  do  a  piece  of  work  one 
half  as  large  ? 

24.  A  man  bought  a  cask  of  wine  containing  63  gal- 
lons, 3  sevenths  of  which  leaked  out;  and  he  sold  the 
remainder  for  36  dollars;  how  much  per  gallon  did  he 
sell  it  for? 

25.  If  a  cask  of  wine  cost  48  dollars,  what  is  5 
eighths  of  it  worth  ? 

26.  A  man  bought  7  oranges  for  6  cents  apiece,  and 
sold  them  all  for  54  cents,  how  much  did  he  gain  by 
the  bargain  ? 

27.  A  man  bought  8  yards  of  cloth  for  56  dollars. 


«EC.  VII.]  ARITHMETIC.  71 

and  sold  it  a<rain  for  9  dollars  a  3^ard ;  how  much  did 
he  gain  by  the  bargain  ? 

28.  A  man  bought  8  barrels  of  flour  for  40  dollars; 
how  much  must  he  sell  it  at  per  barrel  to  gain  16  dol- 
lars ? 

29.  A  man  bought  5  firkins  of  butter  at  7  dollars  a 
firkin ;  how  much  must  he  sell  it  per  firkin  to  gain  10 
dollars  ? 

30.  A  man  gave  35  cents  for  his  breakfast,  which 
was  5  eighths  of  what  he  gave  for  his  dinner;  what  did 
he  give  for  his  dinner  ? 

31.  A  ship's  crew  of  6  men  have  provision  for  3 
months  ;   how  many  months  would  it  last  one  man  ? 

32.  A  ship's  crew  have  provision  sufficient  to  last  1 
man  27  months ;  how  long  would  it  last  9  men  ? 

33.  A  ship's  crew  have  provision  sufficient  to  last  3 
men  10  months ;  how  long  would  it  last  5  men  ? 

34.  A  man  built  40  rods  of  wall  in  a  certain  time ; 
another  man  can  build  9  rods  while  the  first  builds  5; 
how  much  would  he  build  in  the  same  time  ? 


SECTION  VII. 

If  tlie  comhinations  in  this  section  should  he  found 
too  difficulty  they  may  he  omitted  until  reviewing  the 
hooJc. 

A.  1.  A  MAN  being  asked  the  age  of  his  eldest  son, 
answered,  that  his  youngest  son  was  6  years  old,  and 
that  2  thirds  of  the  youngest  son's  age  was  just  1 
fifth  of  the  eldest  son's  age.  Required  the  age  of  the 
eldest  son  ? 

2.  2  thirds  of  6  is  1  fifth  of  what  number? 

3.  A  man  being  asked  how  many  sheep  he  had,  said 
that  he  had  them  in  two  pastures;  in  one  pasture  he 
had  8;  and  that  3  fourths  of  these  was  just  1  third  of 


72  *       ARITHMETIC.  [parti. 

what  he  had  in  the  other.     How  many  were  there  in  the 
other  ? 

4.  3  fourths  of  8  is  1  third  of  what  number  ? 

5.  Two  boys  talking'  of  their  ages,  one  said  he  was  9 
years  old.  Well,  said  the  other,  2  thirds  of  your  age 
is  exactly  3  fourths  of  my  age,  now,  if  you  will  tell  me 
how  old  I  am,  I  will  give  you  as  many  apples  as  I  am 
years  old.     What  was  his  age  ? 

6.  2  thirds  of  9  is  three  fourths  of  what  number? 

7.  Two  boys  counting  their  money,  one  said  he  had 
10  cents.  The  other  says,  4  fifths  of  your  money  is 
exactly  2  sevenths  of  mine ;  now,  if  you  will  tell  how 
many  I  have,  I  will  give  you  1  half  of  them.  How 
many  had  he  ? 

8.  4  fifths  of  10  is  2  sevenths  of  what  number? 

9.  5  sixths  of  12  is  2  thirds  of  what  number  ? 

10.  6  sevenths  of  14  is  4  ninths  of  what  number? 

11.  6  fifths  of  15  is  2  thirds  of  what  number  ? 

12.  7  ninths  of  18  is  2  fifths  of  what  number? 

13.  4  fifths  of  20  is  8  sevenths  of  what  number  ? 

14.  8  sevenths  of  21  is  6  tenths  of  what  number  ? 

15.  5  sixths  of  24  is  10  sevenths  of  how  many  times 
5? 

16.  3  sevenths  of  28  is  2  eighths  of  how  many  times 
7? 

17.  4  fifths  of  30  is  6  sevenths  of  how  many  times 
8? 

18.  6  eighths  of  32  is  8  ninths  of  how  many  times  5? 

19.  4  ninths  of  36  is  8  tenths  of  how  many  times  6? 

20.  3  fourths  of  40  is  5  sevenths  of  how  many  times 
8? 

21.  6  ninths  of  45  is  3  fifths  of  how  many  times  7  ? 

22.  5  sixths  of  48  is  10  sevenths  of  how  many  times 
3? 

23.  4  sevenths  of  63  is  6  fifths  of  how  many  times 
8? 

24.  5  ninths  of  72  is  4  sevenths  of  how  many  times 
9? 


SEC.  VII  ]  ARITHMETIC.  78 

B.  1.  4  fifths  of  15  is  6  tenths  of  how  many  thirds 
of  21? 

2.  4  thirds  of  18  is  8  ninths  of  how  many  sevenths 
of  35  ? 

3.  6  sevenths  of  21  is  2  thirds  of  how  many  thirds  of 
24  i  ^ 

4.  5  fourths  of  24  is  10  sevenths  of  how  many  fifths 
of  40  ? 

5.  5  eighths  of  32  is  2   fifths  of  how  many  fifths 
of  35? 

6.  4  sevenths  of  63  is  6  eighths  of  how  many  ninths 
of  45  ? 

7.  3  sevenths  of  56  is  4  ninths  of  how  many  fourths 
of  28  ? 

8.  3  eighths  of  64  is  6  tenths  of  how  many  sixths 
of  30  ? 

9.  2  eighths  of  72  is  3  tenths  of  how  many  fifths  of 
40? 

C.   1.  Two  times  eleven  are  how  many? 

2.  Two  times  twelve    are  how  many  ? 

3.  Two  times  thirteen  are  how  many? 

4.  Two  times  fourteen  are  how  many  ? 

5.  Two  times  fifteen  are  how  many? 

6.  Two  times  sixteen  are  how  many? 

7.  Two  times  seventeen  are  how  many? 

8.  Two  times  eighteen  are  how  many  ? 

9.  Two- times  nineteen  are  how  many? 

10.  Two  times  twenty  are  how  many? 

11.  Three  times  eleven  are  how  many? 

12.  Three  times  twelve  are  how  many  ? 

13.  Three  times  thirteen  are  how  many? 

14.  Three  times  fourteen  are  how  many? 

15.  Three  times  fifteen  are  how  many? 

16.  Three  times  sixteen  are  how  many? 

17.  Three  times  seventeen  are  how  many? 

18.  Three  times  eighteen  are  how  many? 

19.  Three  times  nineteen  are  how  many? 


ARITHMETIC.  [part  i. 

20.  Three  times  twenty  are  how  many? 

21.  Four  times  eleven  are  how  many  ? 

22.  Four  times  twelve  are  how  many  ? 

23.  Four  times  thirteen  are  how  many  ? 

24.  Four  times  fourteen  are  how  many  ? 

25.  Four  times  fifteen  are  how  many  ? 

26.  Four  times  sixteen  are  how  many  ? 

27.  Four  times  seventeen  are  how  many? 

28.  Four  times  eighteen  are  how  many  ? 

29.  Four  times  nineteen  are  how  many  ? 

30.  Four  times  twenty  are  how  many  ? 

31.  Five  times  eleven  are  how  many? 

32.  Five  times  twelve  are  how  many? 

33.  Five  times  thirteen  are  how  many? 

34.  Five  times  fourteen  are  how  many? 

35.  Five  times  fifteen  are  how  many? 

36.  Five  times  sixteen  are  how  many  ? 

37.  Five  times  seventeen  are  how  many  ? 

38.  Five  times  eighteen  are  how  many? 

39.  Five  times  nineteen  are  how  many?  .•  , 

40.  Five  times  twenty  are  how  many? 

41.  ^ix  times  eleven  are  how  many? 

42.  Six  times  twelve  are  how  many? 

43.  Six  times  thirteen  are  how  many? 

44.  Six  times  fourteen  are  how  many? 

45.  Six  times  fifteen  are  how  many? 

46.  Six  times  sixteen  are  how  many? 

47.  Six  times  seventeen  are  how  many  ? 

48.  Six  times  eighteen  are  how  many? 

49.  Six  times  nineteen  are  how  many  ? 

50.  Six  times  twenty  are  how  many? 

51.  Seven  times  eleven  are  how  many? 

52.  Seven  times  twelve  are  how  many  ? 

53.  Seven  times  thirteen  are  how  many? 

54.  Seven  times  fourteen  are  how  many? 

55.  Seven  times  fifteen  are  how  many  ? 

56.  Seven  times  sixteen  are  how  many  ? 

57.  Seven  times  seventeen  are  how  many? 


SEC.  VII.]  ARITHMETIC.  75 

58.  Seven  times  eighteen  are  how  many? 

59.  Seven  times  nineteen  are  how  many? 

60.  Seven  times  twenty  are  how  many  ? 

61.  Eight  times  eleven  are  how  many? 

62.  Eight  times  twelve  are  how  many? 

63.  Eight  times  thirteen  are  how  many? 

64.  Eight  times  fourteen  are  how  many? 

65.  Eight  times  fifteen  are  how  many? 

66.  Eight  times  sixteen  are  how  many? 

67.  Eight  times  seventeen  are  how  many? 

68.  Eight  times  eighteen  are  how  many? 

69.  Eight  times  nineteen  are  how  many? 

70.  Eight  times  twenty  are  how  many? 

71.  Nine  times  eleven  are  how  many  ? 

72.  Nine  times  twelve  are  how  many  ? 

73.  Nine  times  thirteen  are  how  many  ? 

74.  Nine  times  fourteen  are  how  many? 

75.  Nine  times  fifteen  are  how  many? 

76.  Nine  times  sixteen  are  how  many? 

77.  Nine  times  seventeen  are  how  many? 

78.  Nine  times  eighteen  are  how  many? 

79.  Nine  times  nineteen  are  how  many? 

80.  Nine  times  twenty  are  how  many? 

81.  Ten  times  eleven  are  how  many? 

82.  Ten  times  twelve  are  how  many? 

83.  Ten  times  thirteen  are  how  many? 

84.  Ten  times  fourteen  are  how  many? 

85.  Ten  times  fifteen  are  how  many? 

86.  Ten  times  sixteen  are  how  many? 

87.  Ten  times  seventeen  are  how  many? 

88.  Ten  times  eighteen  are  how  many? 

89.  Ten  times  nineteen  are  how  many? 

90.  Ten  times  twenty  are  how  many? 


76  AlUTHiMETlC.  [parti. 


'     SECTION  YIII. 

A.  1.  If  you  cut  an  apple  into  two  equal  parts,  what 
is  one  of  those  parts  called  ?* 

2.  How  many  halves  of  an  apple  will  make  the 
whole  apple? 

3.  If  you  cut  an  apple  into  3  equal  parts,  what  is 
1  of  those  parts  called?  what  are  2  of  the  parts 
called  ? 

4.  How  many  thirds  of  an  apple  -will  make  the 
whole  apple? 

5.  If  you  cut  an  apple  into  4  equal  parts,  what  is  1 
of  those  parts  called  ?  what  are  2  of  those  parts  called  ? 
what  are  three  of  them  called? 

6.  How  many  fourths  of  an  apple  make  the  whole 
apple  ? 

7.  If  an  apple  be  cut  into  5  equal  parts,  what  is  1 
of  the  parts  called  ?  what  are  2  of  the  parts  called  ? 
what  are  3  of  the  parts  called?  what  are  4  of  the 
parts  called  ? 

8.  How  many  fifths  of  an  apple  make  the  whole 
apple  ? 

9.  If  an  apple  be  cut  into  6  equal  parts,  what  is  1 
of  the  parts  called  ?  what  are  2  of  the  parts  called  ? 
what  3  ?  what  4  ?  what  5  ? 

10.  How  many  sixths  of  an  apple  make  the  whole 
apple  ? 

11.  If  an  apple  be  cut  into  7  equal  parts,  what  is  1 
of  the  parts  called  ?  what  are  2  of  the  parts  called  ? 
what  3  ?  what  4  ?  what  5  ?  what  6  ? 

Let  the  instructor  ask  the  pupils  the  divisions  of  a  unit  in  this  man- 
ner as  far  as  the  division  into  10  parts.  It  would  be  well  to  ask  them 
further.  Then  let  him  begin  again,  and  suppose  an  orange  instead  of 
an  apple.  After  applying  the  division  to  several  different  things,  a 
line  may  be  drawn  on  the  blackboard  and  divided.  It  is  well  often  to 
require  the  pupil  to  make  the  requisite  division  of  a  line  on  the  board. 

12.  A  man  had   a  bushel  of  corn,  and   wished  to 
*  See  Section  III,  article  B,  remark  before  question  1  and  17. 


SEC.  VIII.]  ARITHMETIC.  77 

give  1  half  of  a  bushel  apiece  to  some  laborers;  how 
many  could  he  give  it  to? 

13.  How  many  halves  are  there  in  1  ? 

14.  A  man  divided  2  barrels  of  flour  among  his 
laborers,  giving  them  1  half  of  a  barrel  apieoe;  how 
many  men  did  he  give  it  to? 

15.  How  many  halves  are  there  in  2?* 

16.  In  3  bushels  of  corn,  how  many  half  bushels? 

17.  How  many  halves  are  there  in  3? 

18.  A  boy  divided  4  oranges  among  his  companions, 
giving  them  1  half  of  an  orange  apiece;  how  many  boys 
did  he  give  them  to? 

19.  How  many  halves  are  there  in  4? 

20.  A  man  having  some  laborers,  gave  them  1  half 
a  dollar  apiece;  it  took  3  dollars  and  1  half  a  dollar  to 
pay  them;  how  many  laborers  were  there? 

21.  How  many  halves  are  there  in  3  and  1  half? 

22.  How  many  halves  are  there  in  5? 

23.  How  many  halves  arc  there  in  7  and  1  half? 

24.  How  can  you  tell  how  many  halves  there  are  in 
any  number? 

A71S.  3ince  there  are  2  halves  in  one,  there  must  be 
twice  as  many  halves  as  there  are  whole  ones. 

'25.  If  you  had  1  orange,  and  should  divide  it  among 
your  companions,  giving  them  1  third  apiece,  how 
many  could  you  give  it  to? 

26.  How  many  thirds  are  there  in  1? 

27.  If  you  cut  2  oranges  each  into  3  pieces,  how 
many  pieces  would  they  make  ? 

28.  If  you  cut  3  oranges  into  3  pieces  each,  how 
many  pieces  would  they  make? 

29.  If  you  cut  4  apples  each  into  3  pieces,  how 
many  pieces  would  they  make  ? 

30.  How  many  thirds  are  there  in  2?  in  3?  in  4? 
in  5? 

31.  How  can  you  tell  how  many  thirds  there  are  in 
any  number  ? 

*  See  the  Key. 


78  ARITHMETIC.  [pAitT  i. 

Ans.  Since  there  are  3  thirds  in  one,  there  will  be  3 
times  as  many  thirds  as  there  are  whole  ones. 

32.  If  you  had  2  bushels  and  1  third  of  a  bushel  of 
corn  to  give  to  some  poor  persons,  how  many  could  you 
give  it  to  if  you  should  give  them  1  third  of  a  bushel 
apiece? 

33.  How  many  thirds  are  there  in  2  and  1  third? 

34.  If  a  horse  can  eat  1  third  of  a  bushel  of  oats 
in  1  day,  how  many  days  would  it  take  him  to  eat  3 
bushels  and  2  thirds  of  a  bushel? 

35.  How  many  thirds  are  there  in  3  and  2  thirds? 

36.  If  1  horse  can  eat  1  third  of  a  bushel  of  oats 
in  a  day,  how  many  horses  will  it  take  to  eat  5  bushels 
and  2  thirds  of  a  bushel  in  the  same  time? 

37.  In  5  and  2  thirds  how  many  thirds? 

38.  In  7  and  1  third  how  many  thirds? 

39.  If  a  horse  will  eat  1  fourth  of  a  ton  of  hay  in 
1  month,  how  many  horses  will  eat  a  ton  in  the  same 
time?  How  many  will  eat  2  tons?  How  many  will 
eat  3  tons?     4  tons?     5  tons?     6  tons? 

40.  How  many  fourths  are  there  in  1  ?  in  2  ?  in  3  ? 
in -4?  in  5?  in  6? 

41.  How  can  you  tell  how  many  fourths  there  are  in 
any  number.      '  """ 

42.  How  many  fourths  are  there  in  4  and  1  fourth? 

43.  In  5  tons  of  hay  and  3  fourths  of  a  ton,  how 
many  fourths  of  a  ton  ? 

44.  In  5  and  3  fourths  how  many  fourths  ? 

45.  In  7  and  1  fourth  how  many  fourths? 

46.  In  9  and  3  fourths  how  many  fourths? 

47.  If  a  horse  eat  1  fifth  of  a  ton  of  hay  in  a 
month,  how  many  horses  will  eat  a  ton  in  the  same 
time?  2  tons?  3  tons?  4  tons?  5  tons? 

48.  How  many  fifths  are  there  in  1?  in  2?  in  3? 
in  4?  in  5?  in  7?  in  9? 

49.  How  can  you  tell  how  many  fifths  there  are  in 
any  number? 

50.  In  2  and  1  fifth  how  many  fifths? 


BBC.  vm.]  ARITHMETIC.  79 

51.  In  3  dollars  and  2  fifths  of  a  dollar,  how  many 
fifths  of  a  dollar. 

52.  In  3  and  2  fifths  how  many  fifths? 

53.  In  5  and  3  fifths  how  many  fifths? 

54.  In  G  and  4  fifths  how  many  fifths? 

55.  How  many  sixths  are  there    in  1  ?  in  2?  In  3  ? 
in  4?    in  5?  in  7?  in  8? 

56.  In  2  and  2  sixths  how  many  sixths? 

57.  In  3  and  4  sixths  how  many  sixths  ? 

58.  In  6  and  5  sixths  how  many  sixths  ? 

59.  How  many  sevenths  are  there  in  1?  in  2?  in  3  ? 
in4J  in  6?  in  9?  in  10? 

60.  In  3  and  1  seventh  how  many  sevenths? 

61.  In  5  and  3  sevenths  how  many  sevenths? 

62.  In  7  and  5  sevenths  how  many  sevenths? 

63.  How  many  eighths  are  there  in  1?  in  3  ?  in  5? 
in  8? 

64.  In  2  and  3  eighths  how  many  eighths? 

65.  In  3  and  5  eighths  how  many  eighths  ? 

66.  In  5  and  7  eighths  how  many  eighths  ? 

67.  How  many  ninths  are  there  in  1  ?  in  2  ?  in  7? 
in  5? 

68.  In  2  and  2  ninths  how  many  ninths? 

69.  In  4  and  3  ninths  how  many  ninths? 

70.  In  6  and  4  ninths  how  many  ninths  ? 

71.  In  8  and  7  ninths  how  many  ninths  ? 

72.  How  many  tenths  in  1  ?  in  2  ?  in  5  ?  in  8  ? 

73.  In  3  and  3  tenths  how  many  tenths? 

74.  In  4  and  7  tenths  how  many  tenths? 

75.  In  8  and  9  tenths  how  many  tenths? 

76.  In  7  and  4  tenths  how  many  tenths? 

77.  In  9  and  8  tenths  how  many  tenths? 

78.  In  7  and  4  sevenths  how  many  sevenths? 

79.  In  9  and  2  thirds  how  many  thirds? 

80.  In  10  and  3  fourths  how  many  fourths? 

81.  In  8  and  4  fifths  how  many  fifths? 

82.  In  7  and  5  ninths  how  many  ninths? 


80  ARITHMETIC.  [part  i 

B,  1.  If  you  give  4  men  1  half  of  a  barrel  of  flour 
apiece,  how  many  barrels  will  it  take? 

2.  In  4  halves  how  many  times  1  ? 

3.  If  you  give  3  boys  1  half  of  an  orange  apiece, 
how  many  oranges  will  it  take  ? 

4.  In  3  halves  how  many  times  1  ? 

5.  If  you  give  five  men  1   half  of  a  dollar  apiece,  \ 
how  many  dollars  will  it  take?  k 

6.  In  5  halves  how  many  times  1  ? 

7.  In  6  halves  how  many  times  1  ? 

8.  In  7  halves  how  many  times  1  ? 

9.  How  can  you  tell  how  many  whole  ones  there  are 
in  any  number  of  halves  ? 

10.  A  man  divided  some  corn  among  6  persons, 
giving  them  1  third  of  a  bushel  apiece;  how  many 
bushels  did  it  take  ? 

11.  In  6  thirds  how  many  times  1  ? 

12.  In  5  thirds  how  many  times  1  ? 

13.  A  man  gave  8  paupei:s  1  third  of  a  dollar  apiece, 
how  many  dollars  did  it  take  ? 

14.  In  8  thirds  how  many  times  1  ? 

15.  In  ten  thirds  how  many  times  1  ? 

16.  How  can  you  tell  how  many  whole  ones  there 
are  in  any  number  of  thirds  ? 

17.  If  a  man  spends  1  fourth  of  a  dollar  in  1  day, 
how  many  dollars  will  he  spend  in  8  days?  How  many 
in  7  days?     How  many  in  11  days? 

18.  In  8  fourths  how  many  times  1  ? 

19.  In  7  fourths  how  many  times  1  ? 

20.  In  11  fourths  how  many  times  1? 

21.  In  13  fourths  how  many  times  1  ? 

22.  In  18  fourths  how  many  times  1  ? 

23.  How  can  you  tell  how  many  whole  ones  there  are 
in  any  number  of  fourths  ? 

24.  If  1  fifth  of  a  barrel  of  flour  will  last  a  family 
1  day,  how  many  barrels  will  last  thorn  10  days  ?  How 
many  8  days?    11  days?    17  days? 

25.  In  10  fifths  how  many  times  1  ? 


SEC.  IX.  1  ARITHMETIC.  ^81 

26.  In  8  fifths  how  many  times  1  ? 

27.  In  11  fifths  how  many  times  1  ? 

28.  In  17  fifths  how  many  times  1  ? 

29.  In  18  sixths  how  many  times  1  ? 

30.  In  23  fifths  how  many  times  1  ? 

31.  In  21  sevenths  how  many  times  1? 

32.  In  24ei2;hths  how  many  times  1? 

33.  In  36  ninths  how  many  times  1  ? 

34.  In  30  tenths  how  many  times  1  ? 

35.  In  35  fourths  how  many  times  1  ? 

36.  In  37  eighths  how  many  times  1 '( 

37.  In  43  fifths  how  many  times  1  ? 

38.  In  48  ninths  how  many  times  1  ? 

39.  In  53  tenths  how  many  times  1  ? 

40.  In  57  eighths  how  many  times  1  ? 

41.  In  76  tenths  how  many  times  1  ? 

42.  In  78  ninths  how  many  times  1  ? 


SECTION  IX. 

A.  1.  If  a  breakfast  for  1  man  costs  1  third  of  a  dol- 
lar, what  would  a  breakfast  for  2  men  cost  ? 

2.  How  much  is  2  times  1  third  ? 

3.  If  it  takes  you  1  third  of  an  hour  to  travel  1  mile, 
how  long  will  it  take  you  to  travel  3  miles? 

4.  How  much  is  3  times  1  third? 

5.  If  1  man  can  eat  1  third  of  a  pound  of  meat  at  a 
meal,  how  much  can  5  men  eat  ? 

6^How  much  is  7  times  1  third? 

7v  If  1  man  can  eat  2  thirds  of  a  pound  of  meat 
for  dinner,  how  many  thirds  of  a  pound  would  3  men 
eat? 

8.  How  much  is  2  times  2  thirds  ? 

9.  A  man  gave  to  4  paupers  2  thirds  of  a  dollar 
apiece,  how  many  thirds  of  a  dollar  did  he  give  them  ? 
how  many  dollars? 


82  ARITHMETIC.  [part  i. 

10.5  times  2  thirds  are  how  many  thirds  ?  how  many 
times  1  y 

11.  If  you  give  3  men  1  fourth  of  a  dollar  apiece, 
how  many  fourths  of  a  dollar  will  it  take  ? 

12.  3  times  1  fourth  are  how  many  fourths? 

13*  If  you  give  3  men  3  fourths  of  a  bushel  of  corn 
apiece,  how  many  fourths  of  a  bushel  will  it  take  ?  how 
many  bushels  ? 

14.  5  times  3.  fourths  are  how  many  fourths  ?  how 
many  times  1  ? 

15.  If  1  horse  eats  1  fifth  of  a  bushel  of  oats  in  a  day, 
how  much  will  4  horses  eat  in  the  same  time  ? 

16.  3  times  1  fifth  are  how  many  fifths  ? 

17.  If  1  man  can  earn  3  fifths  of  a  dollar  in  a  day, 
how  much  can  he  earn  in  4  days  ? 

18.  7  times  3  fifths  are  how  many  fifths?  how  many 
times  1  ? 

19.  If  a  family  consume  2  sevenths  of  a  barrel  of 
flour  in  a  week,  how  much  would  they  consume  in  5 
weeks? 

20.  6  times  2  sevenths  are  how  many  sevenths  ?  how 
many  times  1  ? 

21.  5  times  3  eighths  are  how  many  eighths?  how 
many  times  1  ? 

22.  How  much  is  6  times  3  fifths  ? 

23.  How  much  is  7  times  5  sixths  ? 

24.  How  much  is  5  times  4  ninths  ? 

25.  How  much  is  6  times  8  ninths? 

26.  How  much  is  7  times  9  tenths? 

27.  How  much  is  5  times  7  tenths  ? 

28.  How  much  is  6  times  7  eighths?  ^ 

29.  How  much  is  9  times  5  eighths  ?  ^ 

30.  How  much  is  8  times  5  sevenths  ? 

31.  How  much  is  7  times  5  sixths? 

32.  How  much  is  8  times  7  fourths  ? 

33.  How  much  is  7  times  4  fifths  ? 

34.  How  much  is  5  times  3  eighths? 


SEC.  IX.]  ARITHMETIC.  83 

B.  1.  If  1  bushel  of  wheat  costs  a  dollar  and  1  half, 
what  will  2  bushels  cost  ? 

2.  How  much  is  2  times  1  and  1  half?* 

3.  If  a  barrel  of  cider  costs  2  dollars  and  a  half,  what 
will  3  barrels  cost  ? 

4.  How  much  is  4  times  2  and  1  halff 

5.  If  a  barrel  of  beer  costs  3  dollars  and  a  half,  what 
will  2  barrels  cost  ? 

6.  How  much  is  5  times  3  and  1  half? 

7.  How  much  is  6  times  3  and  1  lutlf? 

8.  If  a  box  of  butter  costs  2  dollars  and  1  third  of  a 
dollar,-  what  will  3  boxes  cost  1 

9.  How  much  is  4  times  2  and  1  third? 

10.  If  you  give  to  2  persons  3  bushels  and  a  third 
of  a  bushel  of  wheat  apiece,  how  many  bushels  will  it 
take  ? 

11.  How  much  is  5  times  3  and  1  third? 

12.  If  you  give  to  4  persons  each  2  oranges  and  1 
fourth  of  an  orange,  how  many  oranges  will  it  take? 

13.  How  much  is  5  times  2  and  1  fourth? 

14/  If  it  takes  3  yards  and  2  thirds  of  a  yard  of  cloth 
to  make  a  suit  of  clothes,  how  many  yards  will  it  take 
to  make  2  suits  ? 

15.  How  much  is  4  times  3  and  2  thirds? 

16.  If  a  family  consume  2  bushels  and  2  thirds  of  a 
bushel  of  malt  in  1  month,  how  much  will  they  con- 
sume in  3  months  ? 

17.  How  much  is  5  times  2  and  2  thirds  ? 

18.  How  much  is  4  times  3  and  3  fourths? 

19.  How  much  is  2  times  3  and  1  fourth  ? 

20.  How  much  is  3  times  3  and  3  fourths  ? 

21.  How  much  is  3  times  5  and  1  fourth? 

22.  If  a  horse  eats  3  tons  and  1  fifth  of  a  ton  of  hay 
in  a  year,  how  much  will  2  horses  eat  in  the  same 
time? 

*This  is  to  be  understood  2  times  1  and  2  times  1  half,  and  to  be 
answered  thus :  2  times  1  are  2,  and  two  times  1  half  are  2  halves  or  1, 
which,  added  to  2,  makes  3. 


84  ARITHMETIC.  [parti. 

23.  How  much  is  4  times  3  and  1  fifth  ? 

24.  If  a  man  can  travel  4  miles  and  2  fifths  of  a 
mile  in  1  hour,  how  far  will  he  travel  in  3  hours  ? 

25.  How  much  is  5  times  4  and  2  fifths  ? 

26.  How  much  is  3  times  5  and  Z  fifths  ? 

27.  How  much  is  4  times  6  and  2  fifths? 

28.  How  much  is  3  times  2  and  1  sixth? 

29.  How  much  is  2  times  5  and  3  sixths? 

30.  If  a  yard  of  cloth  costs  4  dollars  and  5  sixths 
of  a  dollar,  what  will  4  yards  cost? 

31.  How  much  is  7  times  4  and  5  sixths? 

32.  How  much  is  2  times  3  and  3  sevenths? 

33.  How  much  is  3  times  4  and  3  sevenths? 

34.  If  a  bushel  of  wheat  costs  7  shillings  and  3 
eighths,  what  will  5  bushels  cost  at  that  rate  ? 

35.  How  much  is  3  times  4  and  5  eigldhs? 

36.  How  much  is  4  times  3  and  7  eighths? 

37.  A  man  bought  8  yards  of  cloth,  at  9  dollars  and 
3  tenths  a  yard;  how  much  did  it  come  to? 

38.  How  much  is  6  times  2  and  2  ninths? 

39.  How  much  is  4  times  5  and  3  ninths? 

40.  A  man  bought  10  barrels  of  cider  at  3  dollars 
and  5  sixths  a  barrel ;  how  much  did  it  come  to? 

41.  How  much  is  3  times  7  and  2  tenths? 

42.  What  cost  8  barrels  of  beef  at  9  dollars  and  3 
sevenths  a  barrel? 

C.  1.  A  boy  wished  to  give  5  other  boys  1  half  pint 
of  chestnuts  apiece;  how  many  pints  would  it  take? 

2.  A  boy  wished  to  give  3  other  boys  3  fourths  of 
an  orange  apiece;  how  many  oranges  would  it  take? 

3.  A  man  gave  to  10  persons  1  fifth  of  a  bushel  of 
corn  apiece;  how  many  bushels  did  it  take? 

4.  A  man  gave  7  men  3  fourths  T)f  a  gallon  of  beer 
apiece;  how  many  gallons  of  beer  did  it  take? 

5.  If  it  takes  1  yard  and  1  fourth  of  a  yard  of  cloth 
to  make  a  pair  of  pantaloons,  how  many  yards  woul(^ 
it  take  to  make  8  pairs  ? 


SEC.  X.]  ARITHMETIC.  85 

6.  If  a  family  consume  2  bushels  and  3  fifths  of  a 
bushel  of  grain  in  1  week,  how  many  bushels  would 
they  consume  in  4  weeks? 

7.  If  a  horse  eats  3  bushels  and  4  sevenths  of  a 
bushel  of  oats  in  1  week,  how  many  bushels  would  he 
eat  in  8  weeks? 

8.  If  a  horse  eats  5  loa<ls  and  3  eights  of  a  load  of 
hay  in  1  year,  how  many  loads  would  6  horses  eat? 

9.  If  a  man  travels  4  miles  and  5  ninths  in  an  hour, 
how  many  miles  would  he  travel  in  8  hours  ? 

10.  If  in  an  orchard  of  10  trees,  each  tree  bears  8 
bushels  and  3  sevenths,  how  many  bushels  will  the 
whole  orchard  bear? 

11.  If  a  man  can  build  5  rods  and  7  eighths  of  a 
rod  of  wall  in  1  day,  how  many  rods  can  he  build  in  8 
days  ? 

12.  If  3  men  can  build  a  piece  of  wall  in  4  days 
and  3  fifths  of  a  day,  how  many  days  would  it  take  1 
man  to  build  it? 

13.  If  1  man  can  build  7  rods  and  2  fifths  of  wall 
in  a  day,  how  many  rods  would  10  men  build? 

•  14.  If  1  man  can  build  3  rods  and  2  ninths  of  wall 
in  1  day,  how  many  rods  would  3  men  build  in  4 
days? 

15.  If  it  takes  1  yard  and  3  sevenths  of  a  yard  of 
cloth  to  make  1  pair  of  pantaloons,  and  2  yards  and  4 
sevenths  for  a  coat ;  how  many  yards  would  it  take  to 
make  3  pairs  of  pantaloons  and  3  coats? 


SECTION  X. 

A.  1.  If  a  yard  of  cloth  costs  3  dollars,  what  will  1 
half  of  a  yard  cost? 

2.  What  is  1  half  of  3  ?* 

3.  If  a  barrel  of  beer  costs  5  dollars,  what  will  1 
half  of  a  barrel  cost  ? 

*  See  Section  V.,  article  A. 


86  ARITHMETIC.  [part  i. 

4.  What  is  1  half  of  5? 

5.  If  2  barrels  of  cider  cost  7  dollars,  what  is  that  a 
barrel  ? 

6.  What  is  1  half  of  7  ? 

7.  What  is  1  half  of  9  ? 

8.  What  is  1  half  of  11? 

9.  What  is  1  half  of  13?    • 

10.  What  is  1  half  of  15? 

11.  If  you  divide  1  bushel  of  wheat  equally  among 
3  persons,  what  part  of  a  bushel  will  you  give  them 
apiece? 

12.  If  3  yards  of  cloth  cost  1  dollar,  what  part  of  a 
dollar  will  1  yard  cost? 

13.  What  is  1  third  of  1? 

14.  How  could  you  divide  2  oranges  into  3  equal 
parts?  that  is,  how  can  you  find  1  third  of  2  oranges?* 

15.  One  third  of  2  oranges  will  be  the  same  as  how 
many  thirds  of  1  orange? 

16.  If  you  divide  2  bushels  of  wheat  equally  among 
3  persons,  what  part  of  a  bushel  will  you  give  them 
apiece  ? 

17.  If  3  bushels  of  corn  cost  2  dollars,  what  part' 
of  a  dollar  will  1  bushel  cost? 

Note. — One  third  of  two  things  is  twice  as  much  as  one  third  of  one 
thing.  One  third  of  one  is  one  third,  and  consequently  one  third 
of  two  things  is  two  thirds.  In  the  same  manner,  one  tJiird  of  four 
things  is  four  thirds  of  one  thing.  If  four  oranges  be  cut  each  into 
three  parts,  and  then  one  part  of  each  be  talven,  it  will  make  four 
pieces,  each  of  which  is  one  third  of  one  orange.  Plence  one  third 
of  four  oranges  is  four  thirds  of  one  orange,  that  is,  one  whole  one 
and  one  third. 

18.  If  3  bushels  of  wheat  cost  4  dollars,  how  much 
is  that  a  bushel? 

19.  What  is  1  third  of  2?  of  4? 

20.  If  3  gallons  of  wine  cost  5  dollars,  what  is  that 
a  gallon  ? 

21.  What  is  1  third  of  5?  of  7?  of  8?  of  10?  of 
11? 

*  Divide  each  orange  into  three  parts,  and  then  take  one  part  from 
each. 


SEC.  X.]  ARITHMETIC.  87 

22.  If  a  bushel  of  apples  be  divided  equally  among 

4  persons,  what  part  of  a  bushel  will  they  have  apiece? 
What  would  they  have  apiece  if  2  bushels  were  divided 
among  them?  What  if  3  bushels?  W^hat  iff)  bush- 
els?    What  if  6  bushels? 

23.  W^hat  is  1  fourth  of  1?  of  2?  of  3?  of  5?  of 
6?  of  7?  of  9?  of  10? 

2-4.  If  a  bushel  of  malt  will  serve  5  persons  1  month, 
how  much  will  serve  1  person  the  same  time? 

25.  If  2  barrels  of  cider  will  serve  5  persons  1 
month,  how  much  will  serve  one  person  the  same  time? 

26.  If  3  barrels  of  flour  be  divided  among  5  men, 
how  much  will  each  have  ?  If  4  barrels  were  divided, 
what  would  each  have  ?  What  if  6  barrels  were 
divided?     What  if  7  barrels  were  divided  ? 

27.  What  is  1  fifth  of  1?  of  2?  of  3?  of  4?  of  6? 
of  7? 

28.  What  is  1  sixth  of  1?  of  2?  of  3?  of  4?  of 
5?  of  7?  of  8?  of  9?  of  10? 

29.  What  is  1  seventh  of  1?  of  2?  of  3  ?  of  4? 
of  6?  of  6? 

30.  What  is  1  eighth  of  1?  of  2  ?  of  3?  of  4?  of 
5?  of  G?  of  7?  of  8?  of  9?  of  10? 

31.  What  is  1  ninth  of  1?  of  2?  of  3?  of  4?  of 
5?  of  6?  of  7?  of  8?  of  9?  of  10?  of  11? 

32.  What  is  1  tenth  of  1  ?  of  2  ?  of  3  ?  of  4  ?  of 

5  ?  of  6  ?  of  7  ?  of  8  ?  of  9  ?  of  10  ?  of  11  ?  of  12  ? 
of  13? 

33.  If  3  yards  of  cloth  cost  2  dollars,  what  will  1 
yard  cost  ?     What  will  2  yards  cost  ? 

34.  If  1  bushel  of  wheat  costs  2  dollars,  what  will  1 
third  of  a  bushel  cost?  What  will  2  thirds  of  a 
bushel  cost? 

35.  What  is  1  third  of  2? 

36.  What  is  2  thirds  of  2? 

37.  If  a  load  of  wood  costs  5  dollars,  what  will  1 
third  of  a  load  cost?  What  will  2  thirds  of  a  load 
cost? 


88  ARITHMETIC.  [part  i. 

38.  What  is  1  third  of  5? 

39.  What  is  2  thirds  of  5? 

40.  What  is  2  thirds  of  7  ? 

41.  If  4  bushels  of  salt  cost  3  dollars,  what  will  1 
bushel  cost  ?     What  will  3  bushels  cost? 

42.  What  is  1  fourth  of  3? 

43.  What  is  3  fourths  of  3? 

44.  If  a  barrel  of  cider  costs  2  dollars,  what  will  1 
fifth  of  a  barrel  cost?  What  will  2  fifths  of  a  barrel 
cost  ? 

45.  What  is  1  fifth  of  2? 

46.  What  is  2  fifths  of  2? 

47.  What  is  1  fifth  of  7? 

48.  What  is  3  fifths  of  7? 

49.  If  7  gallons  of  wine  cost  5  dollars,  what  will  a 
gallon  cost  ?     What  will  4  gallons  cost  ? 

50.  What  is  1  seventh  of  5? 

51.  What  is  4  sevenths  of  5? 

52.  What  is  1  sixth  of  4? 

53.  What  is  5  sixths  of  4?  .      . 

54.  If  you  divide  7  dollars  among  8  men,  what  part 
of  a  dollar  will  you  give  them  apiece?  What  would 
3  of  them  have? 

55.  What  is  1  eighth  of  7? 

56.  What  is  3  eii^hths  of  7? 

57.  What  is  1  ninth  of  10? 

58.  What  is  7  ninths  of  10  ? 

59.  What  is  1  tenth  of  14? 

60.  What  is  4  tenths  of  14? 

61.  If  5  yards  of  cloth  cost  17  dollars,  what  is  that 
a  yard?  What  would  3  yards  cost?  What  would  8 
yards  cost? 

62.  What  is  3  fifths  of  17? 

63.  What  is  8  fifths  of  17? 

64.  What  is  5  sevenths  of  20? 

65.  What  is  2  ninths  of  22? 

66.  What  is  3  eighths  of  27  ? 

67.  What  is  2  thirds  of  28? 


SEC.  X.] 

68. 

69. 
load? 
cost? 

70. 

71. 

72. 

73. 

74. 

75. 

76. 

77. 

78. 

79. 

80. 


ARITHiMETIC. 


89 


What  is  3  fourths  of  31? 

If  5  loads  of  hay  cost  47  dollars,  what  is  that  a 

What  will   2  loads  cost?     What  will   7   loads 

AVhat  will  12  loads  cost? 


-What 
What 
What 
What 
What 
What 
What 
What 
What 
What 
What 


s  2  fifths  of  47  ? 
s  7  fifths  of  47? 
s  12  fifths  of  47? 
s  4  sevenths  of  48? 
s  4  ninths  of  50? 
s  2  sevenths  of  58? 
s  3  eighths  of  61? 
s  4  tenths  of  73  ? 
s  8  ninths  of  6^? 
s  9  tenths  of  78? 
s  7  eighths  of  70? 


B.  1.  If  you  divide  7  apples  equally  between  2  boys, 
how  many  would  you  give  them  apiece? 

2.  How  can  you  divide  5  oranges  eijually  among  3 
persons? 

3.  Divide  3  bushels  of  corn  equally  among  5  men, 
how  much  would  you  give  them  apiece? 

4.  A  boy  had  7  pears,  and  gave  away  3  fifths  of 
them;  how  many  did  he  give  away?  and  how  many 
had  he  left  ? 

5.  If  2  yards  of  cloth  cost  3  dollars,  what  is  that  a 
yard  ? 

6.  If4yardsof  cloth  cost  2  dollars,  what  is  that  a  yard? 
[Let  the  answers  be  given  in  dollars  and  cents,  or  in 

shillings.    1  dollar  is  100  cents.] 

7.  If  5  bushels  of  corn  cost  7  dollars,  what  is  that 
a  bushel? 

8.  If  a  man  receives  8  dollars  for  6  days'  work,  what 
is  that  per  day  ? 

9.  If  3  bushels  of  wheat  cost  8  dollars,  what  will  2 
bushels  cost  ? 

10.  A  man  had  30  dollars,  and  gave  away  3  sevenths 
of  it;  how  much  did  he  give  away? 


90  ARITHMETIC.  [part  i. 

11.  If  4  yards  of  cloth  cost  10  dollars,  what  will  3 
yards  cost? 

12.  If  3  barrels  of  cider  cost  8  dolkrs,  what  will  10 
barrels  cost? 

13.  If  7  pounds  of  flour  cost  40  cents,  what  will  10 
pounds  cost? 

14.  If  4  firkins  of  butter  cost  26  dollars,  what  will 
7  firkins  cost? 

15.  If  3  men  can  do  a  piece  of  work  in  7  days,  how 
long  will  it  take  1  man  to  do  it?  How  long  would  it 
take  4  men? 

16.  If  2  cocks  will  empty  a  cistern  in  3  hours,  in 
how  long  a  time  would  1  empty  it?  In  how  long  a 
time  would  7  cocks  empty  it? 


SECTION  XL 

A.  1.  Bought  1  half  of  a  yard  of  cloth  for  1  dol- 
lar and  1  half ;  what  was  that  a  yard  ? 

2.  1  and  1  half  is  the  half  of  what  number?* 

3.  If  1  third  of  a  yard  of  cloth  costs  1  dollar  and  3 
fourths  of  a  dollar,  how  much  does  a  yard  cost  ? 

4.  2  and  3  fourths  is  1  third  of  what  number? 

5.  If  1  half  of  a  barrel  of  beer  costs  2  dollars  and  1 
fourth  of  a  dollar,  how  much  will  a  barrel  cost  ? 

6.  2  and  3  fourths  is  1  half  of  what  number? 

7.  If  1  fourth  of  a  box  of  lemons  costs  3  dollars  and 
2  thirds  of  a  dollar,  what  will  a  box  cost? 

8.  2  and  2  thirds  is  1  fourth  of  what  number  ? 

9.  3  and  1  half  is  1  fourth  of  what  number? 

10.  If  1  third  of  a  barrel  of  pork  costs  4  dollars  and 
1  third  of  a  dollar,  what  will  a  barrel  cost  ? 

11.  4  and  2  thirds  is  1  third  of  what  number? 

12.  If  1  sixth  of  a  barrel  of  fish  costs  2  dollars  and 
1  fifth  of  a  dollar,  what  will  a  barrel  cost? 

*  See  Section  VI.  article  A.,  and  Section  IX.  article  B. 


SEC.  XI  ]  ARITHMETIC.  91 

13.  3  and  2  fifths  is  1  sixth  of  what  number? 

14.  If  1  fifth  of  a  barrel  of  salmon  costs  3  dollars 
and  2  sevenths  of  a  dollar,  what  is  that  a  barrel? 

15.  4  and  3  sevenths  is  1  fifth  of  what  number? 

16.  If  a  man  can  travel  4  miles  and  o  sevenths  of  a 
mile  in  1  ninth  of  a  day,  how  far  will  he  travel  in  a 
whole  day? 

17.  5  and  3  sevenths  is  1  ninth  of  what  number? 

18.  2  and  4  fifths  is-l  seventh  of  what  number? 

19.  6  and  3  eighths  is  1  eijihth  of  what  number? 

20.  7  and  5  sixths  is  1  third  of  what  number? 

21.  8  and  3  sevenths  is  1  fourth  of  what  number? 

22.  5  and  3  tenths  is  1  seventh  of  what  number? 

23.  8  and  5  ninths  is  1  fifth  of  what  number? 

24.  9  and  4  fifths  is  1  eighth  of  what  number? 

25.  6  and  4  tenths  is  1  tenth  of  what  number? 

26.  7  and  4  ninths  is  1  fifth  of  what  number? 

27.  8  and  7  tenths  is  1  seventh  of  what  number  ? 

B.  1.  If  2  thirds  of  a  barrel  of  beef  cost  3  dollars, 
what  does  1  third  of  a  barrel  cost  ?* 

2.  3  is  2  times  what  number? 

Ans.  3  is  2  times  the  half  of  3  ;  but  1  half  of  3  is 
3  halves,  or  1  and  1  half,!  therefore  3  is  2  times  1  and 
1  half. 

3.  If  3  fifths  of  a  yard  of  cloth  cost  4  dollars,  what 
will  1  fifth  of  a  yard  cost  ? 

4.  5  is  3  times  what  number  ? 

5.  If  3  sevenths  of  a  barrel  of  pork  cost  2  dollars, 
.what  will  1  seventh  of  a  barrel  cost? 

6.  2  is  three  times  what  number? 

7.  If  4  thirds  of  a  bunch  of  shingles  cost  5  dollars, 
what  does  1  third  of  a  bunch  cost? 

8.  7  is  4  times  what  number? 

9.  2  is  4  times  what  number  ? 

10.  A  man  bought  4  ninths  of  a  barrel  of  flour  for 

*  See  Section  VI.,  article  B, 
t  See  Section  X.,  article  A. 


92  ARITHMETIC.  [parti. 

3  dollars ;  what  would  be  the  price  of  1  ninth  of  a  bar- 
rel at  the  same  rate  ? 

11.  5  is  4  times  what  number  ? 

12.  A  man  bought  5  eighths  of  a  hundred  weight  of 
sugar  for  six  dollars ;  what  would  1  eighth  of  a  hun- 
dred weight  cost  at  the  same  rate  ? 

13.  7  is  5  times  what  number  ? 

14.  3  is  5  times  what  number  ? 

15.  8  is  5  times  what  number? 

16.  9  is  4  times  what  number? 

17.  11  is  6  times  what  number  ? 

18.  13  is  7  times  what  number  ? 

19.  14  is  8  times  what  number  ? 

20.  17  is  5  times  what  number  ? 

21.  18  is  8  times  what  number  ? 

22.  17  is  9  times  what  number  ? 

23.  15  is  ten  times  what  number  ? 

24.  20  is  9  times  what  number  ? 

25.  22  is  10  times  what  number  ? 

26.  24  is  7  times  what  number  ? 

27.  If  2  thirds  of  a  barrel  of  beef  cost  8  dollars, 
what  will  1  third  of  a  barrel  cost  ?  What  will  the  whole 
barrel  cost  ? 

28.  *If  5  is  2  thirds  of  some  number,  what  is  1  third 
of  the  same  number?  2  and  1  half  is  1  third  of  what 
number  ?    Then  5  is  2  thirds  of  what  number  ? 

29.  If  3  fourths  of  a  barrel  of  flour  cost  5  dollars, 
what  will  1  fourth  of  a  barrel  cost?  What  will  the  whole 
barrel  cost  ? 

30.  If  8  is  3  fourths  of  some  number,  what  is  1  fourth 
of  the  same  number?  2  and  2  thirds  is  1  fourth  of  what 
number  ?    Then  8  is  3  fourths  of  what  number  ? 

31.  A  man  boucjht  2  sevenths  of  a  barrel  of  oil  for 
5  dollars  j  how  much  will  1  seventh  cost  at  the  same 
rate  ?  how  much  would  a  barrel  cost  ? 

32.  If  9  is  2  sevenths  of  some  number,  what  is  1 


*  See  Sec.  VI.,  Art.  B.,  Examples  8  and  following. 


sec:  XI.]  ARITHMETIC.  93 

seventh  of  the  same  number 't  4  and  1  half  is  1  seventh 
of  what  number  ?  Then  9  is  2  sevenths  of  what  num- 
ber ? 

33.  Boutxht  5  eighths  of  a  chaldron  of  coal  for  7 
dollars  ;  what  is  the  price  of  1  eighth,  at  the  same  rate  ? 
What  is  the  price  of  the  whole  chaldron  't 

34.  If  12  is  5  eighths  of  some  number,  what  is  1 
eighth  of  the  same  number?  2  and  2  fifths  is  1  eighth 
of  what  number?  Then  12  is  5  eighths  of  what  num- 
ber ? 

35.  If  a  man  ejin  do  7  ninths  of  a  piece  of  work  in 
4  days,  how  long  would  it  take  him  to  do  1  ninth  of  it  ? 
how  long  would  it  take  him  to  do  the  whole  ? 

3().  If  3  is  7  ninths  of  some  number,  what  is  1  ninth 
of  the  same  number  ?  3  sevenths  is  1  ninth  of  what 
number?     Then  3  is  7  ninths  of  what  number? 

37.  If  3  fifths  of  a  chaldron  of  coal  cost  8  dollars, 
what  is  the  whole  chaldron  worth  ? 

38.  7  is  3  fifths  of  what  number  ? 

39.  A  man  bought  6  sevenths  of  a  cask  of  raisins 
for  5  dollars  ;  what  was  the  whole  cask  worth  ? 

40.  8  is  6  sevenths  of  what  number  ? 

41.  A  man  had  4  sixths  of  a  week's  board  for  3  dol- 
lars ;  how  much  is  that  for  a  whole  week  ? 

42.  3  is  5  sixths  of  what  number  ? 

43.  9  is  5  eighths  of  what  number? 

44.  10  is  7  thirds  of  what  number? 

45.  11  is  6  fifths  of  what  number  ? 

46.  12  is  7  fifths  of  what  number  ? 

47.  15  is  2  sevenths  of  what  number? 

48.  17  is  3  tenths  of  what  number? 

49.  A  man  bought  5  eighths  of  a  cask  of  wine  for 
19  dollars,  what  would  the  whole  cask  cost  ? 

50.  19  is  3  eighths  of  what  number  ? 

51.  21  is  4  fifths  of  what  number  ? 

52.  Bought  5  ninths  of  a  ton  of  logwood  for  23  dol- 
lars ;  what  would  a  ton  cogt  at  that  rate  ? 

53.  23  is  7  ninths  of  what  number? 


94  ARITHMETIC  [part  i. 

54.  21  is  3  tenths  of  what  number? 

55.  29  is  4  sevenths  of  what  number  ? 

56.  31  is  5  eighths  of  what  number  ? 

57.  33  is  6  sevenths  of  what  number  ? 

58.  38  is  4  ninths  of  what  number  ? 

59.  A  man  bought  1  barrel  and  1  seventh  of  a  bar- 
rel of  wine  for  41  dollars;  what  was  that  a  barrel  ? 

Note. — 1  barrel  and  1  seventh  of  a  barrel  is  the 
salne  as  8  sevenths  of  a  barrel.  If  8  sevenths  of  a 
barrel  cost  41  dollars,  what  does  a  barrel  cost  ? 

60.  35  is  8  sevenths  of  what  number  ? 

61.  A  man  bought  1  ton  and  4  fifths  of  a  ton  (that 
is,  9  fifths  of  a  ton)  of  fustic  for  43  dollars,  what  was 
that  a  ton  ? 

62.  52  is  9  fifths  of  what  number? 

63.  Bought  1  ton  and  2  sevenths  of  a  ton  of  logwood 
for  48  dollars,  what  was  that  a  ton  ? 

64.  67  is  9  sevenths  of  what  number  ? 

65.  53  is  7  tenths  of  what  number  ? 

66.  58  is  9  eighths  of  what  number  ? 

67.  61  is  10  ninths  of  what  number  ? 

68.  Bought  2  barrels  and  1  fifth  of  a  barrel  of  oil 
(that  is,  11  fifths  of  a  barrel)  for  65  dollars;  what  was 
that  a  barrel  ? 

69.  65  is  9  fifths  of  what  number? 

70.  71  is  8  elevenths  of  what  number  ? 

C.  1.  A  boy  gave  away  2  apples  and  1  half,  which 
was  1  fourth  of  all  he  had  ;  how  many  had  he  ? 

2.  A  man  gave  away  3  dollars,  which  was. 2  fifths  of 
all  the  money  he  had ;  how  much  had  he  ? 

3.  A  man  sold  a  cow  for  15  dollars,  which  was  4 
fifths  of  what  she  cost  him ;  how  much  did  he  lose  by 
the  bargain  ? 

4.  A  man  sold  a  piece  of  cloth  for  37  dollars,  which 
was  9  eighths  of  what  it  cost  him;  how  much  did  he 
gain  by  the  bargain  ? 


SEC.  XI.]  ARITHMETIC.  95 

5.  There  is  a  pole  3  fifths  under  water,  and  7  feet 
out  of  the  water  ;   how  Ions:  is  the  pole  ? 

6.  A  man  sold  a  piece  of  cloth  for  47  dollars,  by 
which  bargain  he  lost  2  ninths  of  what  the  cloth  cost 
him  ;  how  much  did  it  cost  him,  and  how  much  did  he 
lose  ? 


MISCELLANEOUS   EXAMPLES. 

1.  If  a  staff  5  feet  lonor  casts  a  shadow  4  feet  at  12 
o'clock,  what  is  the  length  of  a  pole  that  casts  a  shadow 
67  feet  at  the  same  time  ? 

2.  If  58  gallons  of  water,  in  1  hour  run  into  a  cistern 
containing  97  gallons,  and  44  gallons  run  out  in  an  hour, 
in  what  time  will  it  be  filled '/ 

3.  A  mafi  bought  a  cask  of  wine  containing  75  gal- 
lons ;  2  sevenths  of  it  leaked  out,  and  he  sold  the 
remainder  for  1  dollar  a  gallon;  how  much  did  he  sell 
it  for  ? 

4.  A  cask  of  wine  cost  67  dollars;  what  is  5  eighths 
of  it  worth? 

5.  A  man  bought  9  oranges  for  6  cents  and  2  sev- 
enths apiec?e,  and  sold  them  for  67  cents;  what  did  he 
gain  by  the  bargain  ? 

6.  A  man  bought  10  yards  of  broadcloth  for  70 
dollars;  how  must  he  sell  it  per  yard  in  order  to  gain 
14  dollars? 

7.  If,  when  the  days  are  12  hours  long,  a  man  per- 
forms a  journey  in  3  days;  how  many  hours  is  he  in 
performing  it? 

8.  If  a  man  performs  a  journey  in  36  hours,  how 
many  days  would  he  be  in  performing  it  when  the  days 
are  9  hours  long  ? 

9.  If,  when  the  days  are  II  hours  long,  a  man  can 
perform  a  journey  in  5  days,  in  how  many  hours  will 
he  perform  it  ?  In  how  many  days,  when  the  days  are 
9  hours  long  ?  , 


96  ARITHMETIC.  [part  i. 

10.  What  number  added  to  2  fifths  of  33  will  make 
the  number  17  ? 

11.  How  many  yards  of  cloth  that  is  1  quarter  of 
a  yard  wide,  will  line  10  yards  that  is  3  quarters 
wide  ? 

12.  8  yards  of  cloth  that  is  1  quarter  wide,  are 
equal  to  how  many  yards  that  is  4  quarters  wide  ? 

13.  How  many  yards  of  cloth  that  is  3  quarters 
wide,  are  equal  to  7  yards  that  is  5  quarters  wide  ? 

14.  How  many  yards  of  cloth  that  is  6  quarters 
wide,  are  equal  to  37  that  is  4  quarters  wide  ? 

15.  If  a  piece  of  cloth,  5  quarters  wide,  is  worth 
37  dollars,  what  is  a  piece  of  the  same  length,  3  quar- 
ters wide,  worth  ? 

16.  If  cloth  4  quarters  wide  is  worth  8  dollars  a 
yard,  what  is  1  yard  of  the  same  kind  of  cloth,  that  is 
5  quarters  wide,  worth? 


'SECTION  XII.      . 

Parts  of  one  are  called  fractions.  Fractions  may 
be  expressed  by  figures,  as  well  as  whole  numbers.  It 
requires  two  numbers  to  express  a  fraction ;  one  to 
show  into  how  many  parts  one  is  divided,  and  the  other 
to  show  how  many  of  those  parts  are  used.  For  exam- 
ple, if  we  wish  to  express  one  half,  (which  means  that 
one  is  divided  into  two  equal  parts,  and  that  one  part 
is  used,)  we  must  use  the  figure  2  to  express  that  one 
is  divided  into  two  equal  parts  and  the  figure  1  to  show 
that  one  part  is  used.  And  these  must  be  written  in 
such  a  manner  that  we  may  always  know  what  each 
of  them  is  intended  to  express. 

One  half  is  usually  written  thus,  I;  one  number 
above  a  line,  and  the  other  below  it.  The  number 
below  the  line  shows  into  how  many  parts  one  is 
divided,  and  the  number  above  the  line  shows  how 
many  parts  are  used. 


BEC.  XII.]  ARITHMETIC.  97 

One  third  is  written } 

Two  thirds I 

One  fourth J 

Three  fourths f 

Two  fifths I 

Example.  |  of  an  apple  signifies  that  the  apple  is 
to  be  cut  into  7  equal  parts,  and  that  3  parts  are  to  be 
used. 

Illustrate  bj  a  line,  divided  into  7  equal  parts,  and 
three  of  the  parts  taken.  In  the  same  way  illustrate 
the  meaning  of  the  fractions  |,  |,  j^^. 

We  may  observe,  that,  when  one  is  divided  into  3 
parts,  the  parts  are  called  fhit^ds  ;  when  one  is  divided 
into  4  parts,  the  parts  arc  called  fourOu,  etc.;  that  is, 
the  fraction  takes  its  name  from  the  number  of  parts 
into  which  one  is  divided.  The  number  under  the 
line  is  called  the  denominator,  because  it  gives  name 
to  the  fraction  ;  and  the  number  above  the  line  is  called 
the  numerator,  because  it  shows  the  number  of  parts 
used.  Thus  y^,  10  is  the  denominatoTy  and  3  the 
numerator. 

N.  B. — The  pupil  must  be  made  familiar  with  this 
mode  of  expressing  fractions,  and  must  be  able  to 
apply  it  to  any  familiar  objects;  as  apples,  oranges, 
etc.;*  or  by  blackboard,  before  he  is  allowed  to  pro- 
ceed any  farther.  Particular  care  must  be  takfen  to 
make  him  understand  what  the  denominator  signifies, 
and  what  the  numerator,  as  explained  above.  The 
denominator  should  always  be  explained  first. 

The  following  examples  are  a  recapitulation  of  some 
of  the  foregoing  sections,  for  the  purpose  of  showing 
the  application  of  the  above  method  of  writing  frac- 
tions. Having  analyzed  the  question,  the  pupil  may 
write  the  required  fraction  on  the  blackboard. 


*  When  the  numerator  is  larger  than  the  denominator,  the  fraction 
is  called  an  improper  fraction.    See  Key. 
4 


98  ARITHMETIC.  [part  i. 

-'  « 

See  Section  VIII.  A. 

A.  1.  In  2  liow  many  times  J  ?* 

2.  In  3  how  many  times  ^  t 

3.  In  2  how  many  times  i  ? 

4.  In  4  how  many  times  ^  ? 

5.  In  6  how  many  times  \  ? 

6.  In  7  how  many  times  i  ? 

7.  In  8  how  many  times  i  ? 

8.  In  25^"!*  how  many  times  ^  ? 

9.  In  3i  how  many  time  \  ? 

10.  Reduce  4i  to  an  improper  fraction. J 

11.  Reduce  3^  to  an  improper  fraction. 

12.  Reduce  5f  to  an  improper  fraction. 

13.  Reduce  6|  to  an  improper  fraction. 

'  14.  Reduce  8j^^  to  an  improper  fraction. 
15.  Reduce  9|  to  an  improper  fraction. 


B.   1.  I  are  how  many  times  1? 

2.  ^  are  how  many  times  1  ? 

3.  1  are  how  many  times  1? 

4.  I  are  how  many  times  1  ? 

5.  I  are  how  many  times  1  ? 

6.  '^if  are  how  many  times  1  ? 

7.  ^-^  are  how  many  times  1  ? 

8.  ^^^  are  how  many  times  1  ? 
.9.  ^^  are  how  many  times  1? 
10.  f  J  are  how  many  times  1? 


See  Section  IX. 

A.  1.  How  much  is  3  times  ^  ? 
2,  How  much  is  4  times  i? 

*  When  the  numerator  is  lare;er  than  the  denominator,  the  fraction 
ift  called  an  wiproper  fraction.    See  Key. 

+  2}/^  is  read  2  and  1  half.    It  is  called  a  mixed  number. 

I  That  is,  to  find  how  many  fifths  there  are  in  4  and  1  fifth,  first,  and 
how  many  fifths  there  are  in  4. 


SEC.  XII.] 

3. 
4. 
5. 
6. 

7. 

8. 

9. 

10. 


How  much 
How  much 
How  much 
How  much 
How  much 
How  much 
How  much 
How  much 


ARITHMETIC. 

s  3  times  |  ? 
s  4  times  f  ? 
s  5  times  |? 
s  6  times  |? 
s  8  times  j\? 
s  9  times  ^? 
s  10  times  J  ? 
is  9  times  |? 


99 


B.  1.  How  much 

2.  How  much 

3.  How  much 

4.  How  much 

5.  How  much 

6.  How  much 

7.  How  much 

8.  How  much 

9.  How  much 
10.  How  much 


s  3  times  2^? 
s  4  times  3|? 
s  5  times  6|  ? 
s  6  times  4|  ? 
s  7  times  5'|? 
s  8  times  6|  ? 
is  4  times  1(1  §? 
s  9  times  7|  : 
s  8  times  9^  ? 
s  10  times  7|? 


t^^-e  Sections  Y.  &  X. 


1. 

3. 

5. 

7. 

9. 
11. 
13. 
15. 
17. 
19. 


What  is 
What  is 
What  is 
What  is 
What  is 
What  is 
What  is 
What  is 
What  is 
What  is 


of  6? 

df  8?. 

of  9? 

of  14? 

of  5? 

of  7? 
I  of  17? 
I  of  27? 
f  of  47  ? 
I  of  65? 


2. 

4. 

6. 

8. 
10. 
12. 
14. 
16. 
18. 
20. 


What  is  I 
What  is  I 
What  is  J 
What  is  I 
W^hat  i*  ^ 

_3 
7 
3 
7 
4 
15 


What  is 
What  is 
What  is 
What  is 
What  is 


of  6? 
of  9? 
of  10? 
of  5? 
of  7? 
of  35? 
of  26? 
of  37? 
of  42? 
of  75  ? 


See  Sections  VI.  &  XI. 

A.  1.  2  is  i  of  what  number? 
2.  4  is  i  of  what  number  ? 


100  ARITHMETIC.  [part  i. 

3.  8  is  i  of  what  number  ? 

4.  1 J  is  i  of  what  number  ? 

5.  2|  is  i  of  what  number  ? 

6.  4|  is  J  of  what  number  ? 

7.  6|  is  i:  of  what  number? 

8.  7|  is  i  of  what  number  ? 

9.  8|  is  i  of  what  number  ? 
10.  9y^Q  is  J(j  of  what  number  ? 

B.  1.  4  is  f  of  what  number? 

2.  6  is  f  of  what  number  ? 

3.  8  is  I  of  what  number? 

4.  12  is  %  of  what  number  ? 

5.  15  is  I  of,  what  number  ? 

6.  18  is  ^  of  what  number  ? 

7.  20  is  I  of  what  number  ? 

8.  24  is  I  of  what  number  ? 

9.  28  is  J  of  what  number  ? 

10.  30  is  y^5  of  what  number  ? 

11.  3  is  f  of  what  number  ? 

12.  4  is  I  of  what  number? 

13.  5  is  I  of  what  number  ? 

14.  8  is  I  of  what  number? 

15.  9  is  I  of  what  number? 

16.  17  is  J  of  what  number  ? 

17.  25  is  I  of  what  number? 

18.  38  is  f  of  what  number? 

19.  48  is  I  of  what  number  ? 

20.  54  is  y^  of  what  number  ? 


MISCELLANEOUS   EXAMPLES. 

1.  A  man  sold  8  yards  of  cloth   for   3|  dollars    a 
yard;  what  did  it  come  to? 

2.  A  man  sold  a  horse  for  76  dollars,  which  was  |  of 
what  it  cost  him ;  how  much  did  it  cost  him  ? 


SEC.  XIII.]  ARITHMETIC.  101 

3.  A  man  sold  |  of  a  gallon  of  wine  for  40  cents; 
what  was  that  a  gallon  ? 

4.  If  it  will  take  li  yards  of  cloth  to  make  a  coat, 
how  many  yards  will  it  take  to  make  7  coats  ? 

5.  If  1  horse  consumes  o^  bushels  of  oats  in  2  days, 
how  much  would  2  horses  consume  in  5  days  't 

6.  If,  when  the  days  are  Of  hours  long,  a  man  per- 
forms a  journey  in  10  days,  in  how  many  days  would  he 
perform  it  when  the  days  are  12  hours  long  ? 

7.  A  man  sold  8  j'ards  of  cloth  for  7|  dollars  a  yard, 
and  received  8  firkins  of  butter  at  G|  dollars  a  firkin ; 
how  much  was  then  due  to  him  ? 

8.  Two  men  are  o8  miles  apart,  and  are  travelling 
towards  each  other,  one  at  the  rate  of  3  miles  an  hour, 
the  other  2  miles;  how  much  do  they  approach  each 
other  in  an  hour  ?  How  much  in  2  hours?  In  how 
many  hours  will  they  meet?  At  what  distance  from 
each  place  from  which  they  set  out? 


SECTION  XIII. 

A.  1.  If  you  give  ^  of  an  orange  to  oi>e  boy,  and  k 
to  another,  how  much  more  do  you  give  the  first  than 
the  second  ? 

2.  i  of  an  orange  is  how  many  $  of  an  orange  ? 

3.  If  you  give  ^  of  an  orange  to  one  boy,  and  I  to 
another,  how  many  i  would  you  give  away?  How 
many  i  would  you  have  left  ? 

4.  i  and  J  are  how  many  1  ? 

5.  A  man  gave  to  1  laborer  ^  of  a  bushel  of  wheat, 
and  f  to  another;  how  many  i  of 'a  bushel  did  he  give 
to  both  ?    How  many  bushels  ? 

6.  »}  and  f  are  how  many  i  ?    How  many  times  1  ? 

7.  A  man  gave  ^  of  a  barrel  of  flour  to  1  man,  and 
§  of  a  barrel  to  another;  to  which  did  he  give  the 
most  ? 

8.  J  is  how  many  i  ? 


102  ARITHMETIC.  [parti. 

9.  A  man  bought  J  of  a  bushel  of  wheat  at  one  time, 
and  I  of  a  bushel  at  another;  at  which  time  did  he 
buy  the  most  ? 

10.  i  is  how  many  |  ? 

11.  A  man  bought  f  of  a  yard  of  cloth  at  one  time, 
and  I  of  a  yard  at  another ;  at  which  time  did  he  buy 
the  most  ? 

121  t  are  how  many  i  ? 

13.  A  man  wished  to  give  i  of  a  bushel  of  wheat  to 
one  man,  and  i  of  a  bushel  to  another;  but  he  could 
not  tell  how  to  divide  it.  Another  man  standing  by 
advised  him  to  divide  the  whole  bushel  into  six  equal 
parts  first,  and  then  take  i  of  them  for  one,  and  i  of 
them  for  the  other.  How  many  parts  did  he  give  to 
each  ?     How  many  to  both  ?     How  many  had  he  left  ? 

14.  2  is  how  many  I?  ^  is  how  many  i  ?  J  and  i 
are  how  many  i  ? 

15.  A  man  paying  some  money  to  his  laborers,  gave 
each  man  |  of  a  dollar,  and  each  boy  ^  of  a  dollar; 
how  much  more  did  he  give  to  a  man  than  to  a  boy? 

16.  AVhat  is  the  difi'erence  between  J  and  i  ? 

17.  If  a  man  can  earn  |  of  a  dollar  in  a  day,  and  a 
boy  ^  of  a  dollar,  how  much  does  the  man  earn  more 
than  the  boy  ? 

18.  What  is  the  difference  between  |  and  i? 

19.  A  boy  distributing  some  nuts  among  his  com- 
panions, gave  i^  of  a  quart  to  one,  and  ^  of  a  quart  to 
another ;  how  much  more  did  he  give  to  one  than  to 
the  other  ? 

Note. —  Change  them  to  sixths. 

20.  What  is  the  difference  between  i  and  ^  ? 

21.  A  man  having  2  bushels  of  grain  to  distribute 
among  his  laborers,  wished  to  give  ^  of  a  bushel  to  one, 
and  f  of  a  bushel  to  another,  and  the  rest  to  a,  third ; 
but  was  at  a  loss  to  tell  how  to  divide  it ;  at  last  he  con- 
cluded to  divide  each  bushel  into  six  equal  parts,  or 
sixths,  and  then  to  distribute  those  parts.  How  many 
sixths  did  he  give  to  each  ? 


SEC.  XIII.]  ARITHMETIC.  103 

22.  I  is  how  many  i  ? 

23.  A  man  had  a  horse,  and  a  cow,  and  a  sheep. 
The  horse  would  eat  f  of  a -load  of  hay  in  the  win- 
ter, the  cow  ^,  and  the  sheep. i.  How  many  i  of  a 
load  would  each  eat  ?  How  many  i  would  they  all  eat? 
How  many  loads  ? 

24.  A  boy  having  a  quart  of  nuts,  wished  to  divide 
them,,  so  as  to  give  one  companion  i,  another  It.  and  a 
third  i  of  them ;  but  in  order  to  make  a  proper  division, 
he  first  divided  the  whole  into  eight  equal  parts,  and 
then  he  was  able  to  divide  them  as  he  wished.  How 
many  eighths  did  he  give  to  each  ?  How  many  eighths 
had  he  left  for  himself? 

25.  i  is  how  many  i  ?  J  is  how  many  i  ?  ^  and  \ 
and  i  are  how  many  i  ? 

26.  A  man  gave  f  of  a  barrel  of  flour  to  one  man, 
and  I  of  a  barrel  to  another ;  to  which  did  he  give  the 
most  ?     How  much  ? 

27.  Which  is  the  largest,  f  or  |  ?  How  much  the 
largest  ? 

28.  A  boy  having  a  pound  of  almonds,  said  he  in- 
tended giving  i  of  them  to  his  sister,  and  I  to  his 
brother,  and  the  rest  to  his  mamma.  His  mamma, 
smiling,  said  she  did  not  think  he  could  divide  them  so. 

0  yes  I  can,  said  he,  I  will  first  divide  them  into  twelve 
equal  parts,  and  then  I  can  divide  them  well  enough? 
Pray  how  many  twelfths  did  he  give  to  each  ? 

29.  i  is  how  many  J^?  i  is  how  many  J^^  o  and 
i  are  how  many  y^y  ? 

30.  Mr.  Goodman  having  a  pound  of  raisins,  said  he 
would  give  Sarah  J,  and  Mary  1,  and  James  I  of  them, 
and  he  tol4  Charles  he  should  have  the  rest,  if  he  could 
tell  how  to  divide  them.  Well,  said  Charles,  I  would 
first  divide  the  whole  into  twelve  equal  parts,  and  then 

1  could  take  ^  and  J  and  i  of  them.  How  many 
twelfths  would  each  have  ? 

31.  i  and  J  and  ^  are  how  many  yV  ? 

32.  George  bought  a  pine  apple,  and  said  he  would 


104  ARITHMETIC.  [parti. 

give  i  of  it  to  his  papa,  and  |  to  his  mammaj  and  j^^ 
to  his  brother  James,  if  he  could  divide  it.  James 
took  it  and  cut  it  into  twenty  equal  pieces,  and  then 
distributed  them  as  George  had  desired.  How  many 
twentieths  did  he  give  to  each? 

33.  i  is  how  many  ^^  ?  J  is  how  many  -^j^  ?  §  is  how 
many  J^  ?  j^^  is  how  many  -^j^  't 

34.  i  is  how  many  J^? 

35.  ^  is  how  many  jL  ? 

36.  ^  is  how  many  -i  't 

37.  3  is  how  many  -^^  ? 

38.  f  are  how  many  ^  ? 

39.  f  are  how  many  -^j? 

40.  i  is  how  many  y^  r 

41.  I  are  how  many  jK? 

42.  I  are  how  many  j^^  ? 

43.  4  ^^6  how  many  -Jj? 

44.  I  are  how  many  -^^^  ? 

45.  I  are  how  many  g^g  ? 

46.  I  are  how  many  -3'jj  ? 

47.  YO  ^^^^  ^^^  many  ^^^  ? 

48.  lieduce  ^  to  sixths  and  i  to  sixths. 

49.  §  and  |  are  how  many  i  ? 

50.  Reduce  2  ^i^id  1  to  eighths. 

51.  i  and  ^  are  how  many  i? 

52.  i-  an<i  i  are  how  many  I  ? 

53.  f  and  I  are  how  many  }  ? 

54.  i  and' I  are  how  many  i? 

55.  i  and  |  are  how  many  A  ? 

56.  J  and  |  are  how  many  y^j  ? 

57.  '}  and  ^  and  i  are  how  many  i  ? 

58.  '}  and  |  and  Jg  are  how  many  y^^j  ? 

59.  f  and  i-  are  how  many  y^^  ^ 

60.  f  and  i  and  1  are  how  many  j^^? 

61.  I  and  y^  and  i  are. how  many  ^^  ? 

62.  k  and  i  and  f  and  i  and  y^^j  ai^e  how  many  J^  ? 

63.  i  and  -|  are  how  many  j\  ? 

64.  I  and  1  are  how  many  J^? 


SEC.  XIII.]  ARITHMETIC.  105 

65.  f  and  |  are  how  many  J^  ?   * 

66.  i  less  i  are  how  many  i  ? 

67.  t  and  f ,  less  y^^j,  are  how  many  J^  ? 

68.  I  less  I  are  how  many  jL? 

69.  I  less  I  are  how  many  ^  j  ? 

70.  I  less  I  are  how  many  J^? 

71.  I,  and  f,  and  J,  and  -f^,  less  |,  are  how  many 

72.  i,  and  ^,  and  |,  and  -j'y,  and  2*0,  less  i,  are  l^ow 
■many-^i^? 

73.  4  ^"d  6  ^^®  ^^^  many  ^^^  ^ 

74.  f  and  |  are  how  many  g^^? 

75.  I  and  |  are  how  many  g^. 


9 

63  ' 


When  the  denominators  in  two  or  more  fractions  are 
the  same,  the  fractions  are  said  to  have  a  common  de- 
nominator. Thus  I  and  |  have  a  common  denomina- 
tor. We  have  seen  /hat,  when  two  or  more  fractions 
have  a  common  denominator,  they  may  be  added  and 
subtracted  as  well  as  whole  numbers.  We  add  or  sub- 
tract the  numerators,  andwrite  their  sum  or  difference 
over  the  common  denominator.  The  first  part  of  the 
process  in  the  above  examples  was  to  reduce  them  to  a 
common  denominator. 

76.  lleduce  f  and  f  to  a  common  denominator. 
Note. — They  may  be" reduced  to  twelfths. 

If  it  cannot  be  immediately  seen  what  number  m^st 
be  the  common  denominator,  it  may  be  found  by  mul- 
tiplying all  the  denominators  together;  for  that  will 
always  produce  a  number  divisible  by  all  the  denomi- 
nators. 

77.  Reduce  |  and  ^  to  a  common  denominator. 

78.  Reduce  f  and  f  and  |  to  a  common  denomi- 
nator. 

79.  Reduce  ^  and  |  to  a  common  denominator. 

80.  Reduce  |  and  |  to  a  common  denominator. 


106  ARITHMETIC.  [part  i. 

81.  Reduce  }  and  f  and  f  to  a  common  denominator. 

82.  Add  toorether  |  and  ^. 

83.  Add  together  |  and  y^^. 

84.  Add  together  |  and  fg. 

85.  Add  together  |  and  |  and  y^^. 

86.  Subtract  i  from  ^. 

87.  Subtract  j\  from  |.  • 

88.  Subtract  'j  from  f . 

89.  Subtract  |  from  |. 

B.  1.  Mr.  F.  said  he  would  give  ^  of  a  pine  apple 
to  Fanny,  and  |  to  George,  and  the  rest  to  the  one 
that  could  tell  how  to  divide  it,  and  how  much  there 
would  be  left.  But  neither  of  them  could  tell;  so  he 
kept  it  himself.  Could  you  have  told  if  you  had  been 
there?  How  would  you  divide  it?  How  much  would 
be  left? 

2.  A  man  sold  li  bushel  of  wheat  to  one  man,  4f 
bushels  to  another;  how  many  bushels  did  he  sell  to 
both? 

3.  A  man  bought  6  J  bushels  of  wheat  at  one  time, 
and  2 J  at  another;  how  much  did  he  buy  in  the 
whole? 

4.  A  man  bought  7t  yards  of  one  kind  of  cloth, 
and  6f  yards  of  another  kind;  how  many  yards  in  the 
whole  ? 

5.  A  man  bought  |  of  a  barrel  of  flour  at  one  time, 
2^  barrels  at  another,  and  6|  at  another;  how  much 
did  he  buy  in  the  whole? 

.  6.  A   man   bought   one   sheep  for  4|  dollars,  and 
another  for  5|  dollars;  how  much  did  he  give  for  both? 

7.  There  is  a  pole  standing,  so  that  ^  of  it  is  in  the 
mud,  and  |  of  it  in  the  water,  and  the  rest  out  of  the 
water;  how  much  of  it  is  out  of  the  water? 

8.  A  man  having  undertaken  to  do  a  piece  of  work, 
did  i  of  it  the  first  day,  i  of  it  the  second  day,  and  i 
of  it  the  third  day;  how  much  of  it  did  he  do  in  three 
days? 


SBC.  xin  ]  ARITHMETIC.  107 

9.  A  man  having  a  piece  of  work  to  do,  hired  two 
men  and  a  boy  to  do  it.  The  first  man  could  do  ^  of 
the  work  in  a  day,  and  the  other  i  of  it,  and  the  boy 
I  of  it;  how  much  of  it  would  they  all  do  in  a  day? 

Note. — By  dividing  a  line  into  halves,  and  then  into 
fourths,  it  will  be  seen  that |  is  the  same  as  ^j  a  line 
divided  into  halves  and  then  into  sixths,  will  show  that 
I  is  the  same  as  i,  and  |  as  |j  f>  i>  can  therefore  be 
reduced  to  J,  and  ^  to  f ,  This  is  called  reducing  frac- 
tions to  their  lowest  terms.  It  is  done  by  dividing  the 
greatest  number  that  will  divide  it  without  a  remainder. 

1.  Reduce  |  to  its  lowest  terms.*     Aiis.  f . 

2.  Reduce  -^^  to  its  lowest  terms. 

3.  Reduce  |  to  its  lowest  terms, 

4.  Reduce  y*^  to  its  lowest  terms. 

5.  Reduce  jj  to  its  lowest  terms. 

6.  Reduce  f^  to  its  lowest  terms. 

7.  Reduce  ^q  to  its  lowest  terms. 

8.  Reduce  ^|  to  its  lowest  terms. 

9.  Reduce  J  4  to  its  lowest  terms. 

10.  Reduce  ^^^  to  its  lowest  terms. 

11.  Reduce  ||  to  its  lowest  terms. 

12.  Reduce  |i  to  its  lowest  terms. 

13.  Reduce  ||    to  its  lowest  terms. 

14.  Reduce  |j  to  its  lowest  terms. 

Note. — It  will  be  seen  by  the  above  section  that  if 
both  the  numerator  and  denominator  be  multiplied  by 
the  same  number,  the  value  of  the  fraction  will  not  be 
altered;  or  if  they  can  both  be  divided  by  the  same 
number  without  a  remainder,  the  fraction  will  not  be 
altered. 


*  If  this  article  should  be  found  too  difficult  for  the  pupil,  he  may 
omit  it  till  after  the  next  section. 


108  ARITHMETIC.  [part  i. 

SECTION  XIV. 

A.  1.  A  BOY  having  i  of  an  orange  gave  away^  of 
that,  what  part  of  the  whole  orange  did  he  give  away? 

2.  What  is  i  of  i? 

3.  If  you  €ut  an  apple  into  three  pieces,  and  then 
cut  each  of  those  pieces  into  two  pieces,  how  many 
pieces  will  the  whole  apple  be  cut  into?  What  part 
of  the  whole  apple  will  one  of  the  pieces  be  ? 

4.  What  is  J  of  i? 

5.  A  boy  had  J  of  a  pine  apple,  and  cut  that  half 
into  three  pieces,  in  order  to  give  away  i  of  it.  What 
part  of  the  whole  apple  did  he  give  away? 

6.  What  is  i  of  ^?  ^ 

7.  If  an  orange  be  cut  into  4  parts,  and  then  each 
of  the  parts  be  cut  in  two,  how  many  pieces  will  the 
whole  be  cut  into? 

8.  What  is  -J  of  i? 

9.  A  man  having  }  of  a  barrpl  of  flour,  sold  i  of 
that;  how  much  did  he  sell? 

10.  What  is  i  of  *? 

11.  If  an  orange  be  cut  into  4  equal  parts,  and  each" 
of  those  parts  be  cut  into  3  equal  parts;  how  many 
parts  will  the  whole  orange  be  cut  into  ? 

12.  What  is  iof  i? 

13.  A  boy  having  i  of  a  quart  of  chestnuts,  gave 
away  I  of  what  he  had ;  what  part  of  the  whole  quart 
did  he  give  away? 

14.  What  is  i  of  i? 

15.  What  is  Jof  J?      • 

16.  A  man  owning  ^  of  a  ship,  sold  J  of  his  share; 
what  part  of  the  ship  did  he  sell,  and  what  part  did 
he  then  own  ? 

n.  What  is  i  of  1? 

18.  What  is  J  of  i  ? 

19.  What  is  i  of  i? 

20.  What  is  i  of  ^? 

21.  What  is  i  of  ^? 


SEC.  XIV.] 


ARITHMETIC. 


109 


^ofi? 


s  i  of  4  ? 
siof  i? 
s  J  of  i? 
s  i  of  \? 
s  i  of  J? 
s^of  Vr' 


22.  What 

23.  What 

24.  What 

25.  What 

26.  What 

27.  What 

28.  What 

29.  A  boy  having  f  of  an  orange,  (that  is,  2  pieces,) 
gave  his  sister  ^  of  what  he  had ;  how  many  thirds 
did  he  give  her  ? 

30.  What  is  iof  t?  ' 

31.  A  boy  having  f  of  a  pine  apple,  said  he  would 
give  one  half  of  what  he  had  to  his  sister,  if  she  could 
tell  him  how  to  divide  it.  His  sister  says,  you. have 
I,  or  three  pieces,  if  you  cut  them  all  in  two,  you 
can  give  me  ^  of  them.  But  J  of  J  is  i,  therefore  I 
shall  have  |  of  the  whole  pine  apple. 

32.  What  is  I  of  f  ? 

33.  A  man  owning  f  of  a  share  in  the  Planters' 
Bank,  sold  i  of  his  part ;  what  part  of  a  share  did 
he  sell? 

34.  What  is  i  of  f  ? 

35.  A  man  owning  |  of  a  ship,  sold  k  of  his  share; 
what  part  of  the  whole  ship  did  he  sell?  What  part 
had  he  left? 


36. 

What  ] 

IS  i  of  1? 

37. 

What 

IS  *  of  1? 

38. 

What 

IS  i  of  4  ? 

39. 

What 

IS  i  of  9  ? 

40. 

What 

Is  \  of  V  ? 

41. 

What] 

S   ^  of    3  ? 

42.  A  man  owning  §  of  a  share  in  the  Planters' 
Bank,  sold  i  of  his  part;  what  part  of  a  whole  share 
did  he  sell  ? 

43.  What  is  J  of  I?  '  • 

44.  What  is  i  of  f 

45.  A  boy  having  |  of  a  watermelon,  wished  to 
divide  his  part  equally  between  his  sister,  his  brother, 


110 


ARITHMETIC. 


[part  I. 


and  himself,  but  was  at  a  loss  to  know  how  to  do  it ; 
but  his  sister  advised  him  to  cut  each  of  the  fifths  into 
3  equal  parts.  How  many  pieces  did  each  have  ?  and 
what  part  of  the  whole  melon  was  each  piece? 


46 
47. 

48. 


51. 
52. 
53. 
54. 
55. 
56. 


What 
What 
What 

49.  What 

50.  What 
What 
What 
What 
What 
What 
What 

57.  What 

58.  What 
What 
What 
What 
What 
What 
What 
What 
What 
What 
What 


59. 
60. 
61. 
62. 
63. 
64. 
65. 
66. 
67. 
68. 


iof  I 
lof  f? 
iof  1? 
iof  f? 
T^uOf  I? 

s  ^i  of  i  ? 
s  I  of  i? 
si  of  I? 
sf  of  I? 
s  i  of  §  ? 
s  I  of  I ? 
si  of  I? 
slof  I?' 
s  |of  f? 
s  f  of  I ? 
sf  of  I? 
s^of  f? 
s|of.|? 

s     1     of   t? 
s  y  Q  or    8  : 

S   JO    01     8  i 
S   I   of   y3_? 

s  ^  of  ^  ^ 
sf  of  f? 


69.  If  a  yard  of  cloth  cost  2j  dollars,  what  will  i 
of  a  yard  cost? 

70.  What  is  ^  of  2i? 

71.  A  boy  had  2^  oranges,  and  wished  to  give  i  of 
them  to  his  sister,  and  2'  to  his  brother,  but  he  did  not 
know  how  to  divide  them  equally.  His  brother  told 
him  to  cut  the  whole  into  halves,  and  then  cut  each  of 
the  halves  into  3  pieces.  What  part  of  a  whole  orange 
did  each  have? 

72.  What  is  i  of  2i? 


fiEC.  XIV.]  ARITHMETIC.  Ill 

73.  A  man  bought  4  bushels  of  corn  for  3f  dollars; 
what  part  of  a  dollar  did  1  bushel  cost? 

Change  the  3f  to  thirds,  and  then  Jind  k  of  ^  as 
above, 

74.  What  is  iof  5§? 

75.  If  5  bushels  of  wheat  cost  71  dollars,  what  is 
that  a  bushel  ? 

76.  What  is  -Jof  7f? 

77.  A  man  bought  6  gallons  of  alcohol  for  8|  dol- 
lars;  what  was  that  a  gallon? 

78.  What  is  i  of  8§  ? 

79.  A  man  bought  7  gallons  of  wine  for  8|  dollars; 
how  much  was  that  a  gallon? 

80.  What  is  4  of  8|? 

81.  A  man  bought  10  pieces  of  nankin  for  6|  dol- 
lars; how  much  was  it  a  piece? 

82.  What  is  J^of  6|? 

83.  If  9  bushels  of  rye  cost  7|  dollars,  what  is  that 
a  bushel? 

84.  What  is  1  of  7f? 

85.  What  is  ^  of  5|? 

86.  What  is  i  of  8|? 

87.  What  is  \  of  6-^-3^? 

88.  What  is  i  of  94? 

89.  A  man  bought  7  yards  of  cloth  for  18|  dollars; 
what  was  that  a  yard?  What  would  3  yards  cost  at 
that  rate? 

90.  What  is  4  of  18f.?     What  is  -|  of  18 j? 

91.  A  man  bought  5  barrels  of  cider  for  27|  dollars; 
what  was  it  a  barrel?  What  would  7  barrels  cost  at 
that  rate? 

92.  What  is  \  of  271     What  is  f  of  27|? 

93.  If  6  barrels  of  flour  cost  38|  dollars,  what  would 
10  barrels  cost  at  that  rate  ? 

94.  What  is  i^o  of  38|? 

3.  1.  A  man  bought  a  piece  of  cloth  for  42|  dollars, 


112  ARITHMETIC.  [paet  i. 

and  was  obliged  to  sell  it  for  |  of  wliat  it  cost  Mm; 
how  much  did  he  lose? 

2.  A  man  bought  a  quantity  of  flour  for  53f  dol- 
lars, and  sold  it  for  |  of  what  it  cost  him;  how  much 
did  he  gain? 

3.  If  7  men  can  do  a  piece  of  work  in  4|  days, 
how  long  will  it  take  1  man  to  do  it?  How  long  will 
it  take  3  men  to  do  it  ? 

4.  If  4  men  can  do  a  piece  of  work  in  9|  days, 
how  long  would  it  take  to  do  it  if  7  men  were  em- 
ployed ? 

5.  There  is  a  pole  standing  so  that  |  of  it  is  in  the 
water,  and  |  as  much  in  the  mud;  how  much  is  in  the 
mud? 

6.  If  a  man  can  travel  13|  miles  in  3  hours,  how 
many  miles  can  he  travel  in  8  hours? 

7.  If  5  horses  will  eat  26f  loads  of  hay  in  a  year, 
what  will  8  horses  eat  in  the  same  time? 

8.  If  4  cocks  will  empty  a  cistern  in  6|  hours,  how 
long  will  it  take  7  cocks  of  the  same  size  to  empty  it? 


SECTION  XV. 

A.  1.  A  BOY  having  2  oranges  wished  to  give  }  of 
an  orange  apiece  to  his  playmates;  how  many  could  he 
give  them  to?  If  he  had  given  f  of  an  orange  apiece, 
how  many  could  he  have  given  them  to? 

2.  How  many  times  i  are  there  in  2?  How  many 
times  f  are  there  in  2? 

3.  A  man  having  3  bushels  of  corn,  distributed  it 
among  some  poor  persons,  giving  them  |  of  a  bushel 
each;   to  how  many  did  he  give  it? 

Note. — First  Jind  Jwiv  many  lie,  would  have  given  it 
to,  if  he  had  given  \  of  a,  bushel  to  each. 

4.  In  3  are  how  many  times  i  ?    How  many  times  f  ? 

5.  If  I  of  a  barrel  of  flour  will  last  a  family  one 


SEC.  XV.]  ARITHMETIC.  113 

month,  how  long  will  4  barrels  last  the  same  family? 
How  long  will  6  barrels  last?  How  long  will  10  bar- 
rels last? 

6.  How  many  times  is  |  contained  in  4?  How 
many  times  in  G  ?     How  many  times  in  10  ? 

7.  If  4  of  a  bushel  of  wheat  will  last  a  family  one 
week,  how  many  Weeks  will  6f  bushels  last  the  same 
family? 

8.  How  many  times  is  f  contaiacd  in  6|? 

9.  There  is  a  cistern  havingr  a  cock  which  will  fill 
it  in  I  of  an  hour;  how  mariy  times  would  the  cock 
fill  the  cistern  in  3|  hours? 

10.  How  many  times  is  |  contained  in  3|? 

11.  How  much  cloth  at  1?  dollars  (that  is  |  dollars) 
a  yard,  can  be  -bought  for  4  dollars  ? 

12.  How  many  times  is  1?  or  |  contained  in  4? 

13.  A  man  distributed  8^  bushels  of  wheat  among 
some  poor  persons,  giving  1^  bushels  to  each;  how 
many  did^he  give  it  to? 

14.  How  many  times  is  1^  contained  in  8^? 

15.  If  a  soldier  is  allowed  1^^  pound  (that  is  |  of  a 
pound)  of  meat  in  a  day,  to  how  many  soldiers  would 
6|  pounds  be  allowed? 

16.  How  many  times  is  IJ  contained  in  6|? 

17.  If  If  tons  of  hay  will  keep  a  horse  through  the 
winter,  how  many  horses  will  10  tons  keep? 

18.  How  many  times  is  If  contained  in  10? 

19.  At  2i  dollars  a  box,  how  many  boxes  of  raisins 
can  be  bought  for  10  dollars? 

20.  How  many  times  is  2^  contained  in  10? 

21.  At  1|  dollars  a  pound,  how  many  pounds  of 
indigo  can  be  bought  for  9|  dollars? 

22.  How  many  times  is  1|  contained  in  9|? 

23.  At  1|  dollar  a  barrel,  how  many  barrels  of 
raisins  can  be  bought  for  9|  dollars  ? 

24.  How  many  times  is  14  contained  in  9|? 

25.  At  I  of  a  dollar  apiece,  how  many  pieces  of 
nankin  can  be  bought  for  8|  dollars? 


114  ARITHMETIC.  [paet  i. 

26.  How  many  times  is  i  contained  in  8|? 

27.  At  ^  of  a  dollar  a  pound,  how  many  pounds  of 
tea  can  be  bought  for  7|  dollars? 

28.  How  many  times  is  |  contained  in  7|  ? 

29.  How  many  times  is  3^  contained  in  7f  ? 
80.  How  many  times  is  5  J  contained  in  17  ? 

31.  How  many  times  is  4i  contained  in  9|? 

32.  How  many  times  is  3|  contained  in  12^? 

B.  1.  At  J^  of  a  dollar  a  pound,  how  many  pounds 
of  meat  can  be  bought  for  i^  of  a  dollar? 
Note. —  Change  ^  to  tenths. 

2.  How  many  times  is  ^^  contained  in  J? 

3.  A  man  having  f  of  a  barrel  of  flour,  distributed 
it  among  some  poor  persons,  giving  them  i  of  a  barrel 
apiece;  how  many  did  he  give  it  to? 

Note. —  Change  both  fractions  to  twelfths;  that  is, 
reduce  them  to  a  common  denominator. 

4.  How  many  tim.es  is  i  contained  in  f  ?    , 

5.  If  a  pound  of  almonds  cost  i  of  a  dollar,  how 
many  pounds  can  be  bought  for  f  of  a  dollar  ? 

Note. — Reduce  the  fractions  to  a  common  denomi- 
nator. 

6.  How  many  times  is  ^  contained  in  f  ? 

7.  If  a  piece  of  nankin  cost  |  of  a  dollar,  how  many 
pieces  can  be  bought  for  4f  dollars?  that  is,  ^9  dollars? 

8.  How  many  times  is  |  contained  in  4f  ? 

9.  If  a  bushel  of  barley  cost  |  of  a  dollar,  how 
many  bushels  can  be  bought  for  i  of  a  dollar?  How 
many  for  If  of  a  dollar. 

10.  How  many  times  is  |  contained  in  |^?  How 
many  times  in  1|  ? 

11.  How  many  times  is  |  contained  in  |  ? 

12.  How  many  times  is  |  contained  in  |  ? 


PARTLJ  ARITHMETIC.  115 

TABLES  OF  COINS,  WEIGHTS,  AND  MEASURES. 
Table  1. — Federal  Money. 

Federal  Money  is  the  currency  of  the  Confederate  States,  and 
of  the  United  States. 

10  mills  (m.)  make  1  cent,  marked  cf. 

10  cents  "        1  dime,  "        d. 

10  dimes  "       1  dollar,  "        doll,  or  $. 

10  dollars  "       1  Eagle,  "        E. 

Table  II. — Sterling  Monet. 

Sterling  Money  is  the  currency  of  England. 

4  farthings  (qr.  or  far.)  make  1  penny,  marked  d 

12  pence  "      1  shilling,  "        s. 

20  shillings  "      1  pound,  or  a  sovereign,  "       £,. 

21  shillings  "      1  guinea. 

Table  III— Troy  Weight. 

Used  in  weighing  Gold,  Silver,  and  some  Liquids. 

24  grains  (gr.)  make  1  pennj-weight,  marked  dwt. 
20  pennyweights  "  1  ounce,  "  oz. 
12  ounces                                  "      1  pound.                                 "  lb. 

Table  /,F.— Avoirdupois  Weight. 

Used  in  weighing  Groceries,  Hay,  etc.,  and  all  the  Metals, 
except  Gold  and  Silver.  It  is  now  the  custom  to  allow  100 
lbs.  for  a  hundred  weight,  instead  of  112  as  formerly. 

16  drams  (dr.)  make  1  ounce,  marked  oz. 

16  ounces  "      1  pound,  '         "         lb. 

25  pounds  "       1  quarter,  "        qr. 
4  quarters  "      1  hundred  weight,  "      cwt 

20  hundred  weight  "      1  ton.  "         T. 

Table  V. — Cloth  Measure. 
Used  in  measuring  Woollens  and  other  Cloths. 

214  inches  (in.)  make  1  nail,  marked  na. 

4  nails,  or  9  in.  «      1  quarter  of  a  yard,  "         qr. 

4  quarters  «      1  yard,  "         yd. 

3  quarters  «      1  Flemish  ell,  «         Fl.  e. 

5  quarters  "      1  English  ell,  "         E.  e. 

6  quarters  «      1  French  ell.  "         F.  e. 


116  ARITHMETIC.  [pakt  i. 

Table  VL — Wine  Measure. 
Used  in  measuring  "Wines,  etc. 

4  giUs  (gi.)  make  1  pint,                          marked  pt. 

2  pints  "  1  quart,  "■         qt. 

4  quarts  "  1  gallon,  "          gal. 

5VA  gallons  "  1  barrel,  "  bar.  or  bbl. 

42  gallons  "  1  tierce,  "          tier. 

63  gallons,  or  2  barrels  "  1  hogshead,  "         hhd. 

2  hogsheads  "  1  pipe,  or  butt,  "          pi. 

2  pipes  "  1  tun.  "          tun. 

Table  VII. — Dry  Measure. 

Used  in  measuring  Grains,  Salt,  Oysters,  etc. 

2  pints  make  1  quart,  marked  qt. 

8  quarts  "      1  peek,  "        pk. 

4  pecks  "      1  bushel.  "        bu 

Table  VIII. — Measure  of  Time. 

60  seconds  (sec.)  make  1  minute,  marked  min. 

60  minutes  "  1  hour,  "  hr. 

24  hours  "  1  day,  "  d;/. 

7  days  "  1  week,  "  wk. 

4  weeks  "  1  month,  "  mo. 

13  months.  1  day  and  6  hours,  I «  ^         j.  «  ,^ 

or  12  calendar  months,  J 

For  convenience  of  reckoning,  it  is  usual  in  calen- 
dars to  call  the  year  365  days  for  3  successive  years, 
and  every  fourth  year  366,  (for  in  4  years,  the  six 
hours  overplus  amount  to  a  day,)  which  is  called 
bissextile,  or  leap  year.  This  day  is  added  to  February. 

The  common  year  is  divided  into  twelve  months, 
which  are  sometimes  called  calendar  months,  because 
they  are  the  months  used  in  calendars. 

The  names  of  the  months,  and  the  number  of  days 
in  each,  are  as  follows  : 

Names.  Number  of  days. 

January  31 

February  28,  in  leap  year  29. 

March  31 

April  30 

May  31 


PART  I]  ARITHMETIC.  117 

Names.  Number  of  days. 

June  30 

July  31 

August  31 

September  30 

October  3l 

November  30 

December  31 


MISCELLANEOUS   EXAMPLES. 

1.  In  2  pounds  how  many  ounces? 

2.  In  8  yards  how  many  quarters? 

3.  In  3  quarters  of  a  yard,  how  many  nails  ? 

4.  -A  of  a  dollar  is  how  many  cents  ? 

5.  How  many  farthings  is  |  of  a  penny? 

6.  How  many  pence  is  |  of  a  shilling? 

7.  I  of  a  yard  is  how  many  quarters  and  nails? 

8.  In  £|  how  many  shillings  ? 

9.  How  much  is  |  of  a  shilling  ? 

10.  How  much  is  |  of  a  bushel  of  wheat  ? 

11.  How  much  would  |  of  a  barrel  of  wine  cost,  at 
one  dollar  a  gallon  ? 

12.  How  much  would  i  cwt.  of  sugar  cost,  at  8  cents 
a  pound  ? 

13.  How  much  is  f  of  a  day? 

14.  How  much  is  |  of  a  day  ? 

15.  How  much  is  |  of  a  week  ? 

16.  How  much  is  f  of  an  hour  ? 

17.  How  much  would  |  of  a  hogshead  of  wine  cost, 
at  2  dollars  a  gallon  ? 

18.  If  a  man%pends  28  dollars  in  a  month,  what  is 
that  a  week  ?     How  much  a  day  ? 

19.  If  a  man  spends  16  dollars  a  week,  what  is  that 
a  day  ? 

20.  If  a  man  buys  4  bushels  of  grain  for  5  dollars, 
how  much  is  that  a  bushel  ? 


118  ARITHMETIC.  [part  i. 

21.  If  wine  is  2  dollars  a  gallon,  how  much  is  that  a 
pint  ? 

22.  If  you  give  5  cents  a  gill  for  wine,  what  is  that 
a  pint  y     What  is  it  a  quart  ?     What  is  it  a  gallon  ? 

28.  If  wine  is  worth  20  cents  a  pint,  what  is  that  a 
gill  ?     What  is  it  a  quart  ?     What  is  it  a  gallon  ? 

24.  If  a  yard  of  cloth  is  worth  7  dollars,  what  are 
2 1  'yards  worth  ?  , 

2o.  If  a  man  earns  11  dollars  a  week,  what  is  that  a 

day?  What  for  3  days  't     What  for  4^  days  ? 

2o.  If  a  man  earns  2|  dollars  in  a  day,  what  will  ho 
earn  in  a  week  ? 

27.  What  is  |  of  a  hogshead  of  wine? 

2^.  1  farthing  is  what  part  of  a  penny  ? 

29.  2  fjirthings  is  what  part  of  a  pennyf 

30.  3  farthings  is  what  part  of  a  penny? 

31.  1  penny  is  what  part  of  a  shilling  ? 

32.  2'  pence  is  what  part  of  a  shilling  ? 

33.  3  pence  is  what  part  of  a  shilling? 

34.  5  pence  is  what  part  of  a  shilling  ? 
85.  6  pence  is  what  part  of  a  shilling? 

36.  7  pence  is  what  part  of  a  shilling  ? 

37.  8  pence  is  what* part  of  a  shilling? 

38.  9  pence  is  what  part  of  a  shilling? 

39.  10  pence  is  what  part  of  a  shilling? 

40.  11  pence  is  what  part  of  a  shilling? 

41.  1  shilling  is  what  part  of  a  pound? 

42.  2  shillings  is  what  part  of  a  pound? 

43.  3  shillings  is  what  part  of  a  pound? 

44.  4  shillings  is  what  part  of  a  pound? 

45.  5  shillings  is  what  part  of  a  pound  ? 

46.  What  part  of  a  pound  is  6  shillings?  7  shil- 
lings? 8  shillings?  9  shillings?  10  shillings?  11  shil- 
lings? 12  shillings?  13  shillings?  14  shillings?  15 
shillings?  16  shillings?  17  shillings?  18  shillings? 
19  shillings? 

47.  How  many  farthings  are  there  in  a  shilling  ? 

48.  One  farthing  i%  what  part  of  a  shilling? 


PART  I.]  ARITHMETIC.  119 

49.  2  farthings  is  what  part  of  a  shilling?  3  far- 
things? 4  farthings?  5  farthings?  6  farthings?  7 
farthings?    8  fafthmgs?    9  farthings?    10  farthings? 

50.  How  many  pence  are  there  in  a  pound?    . 

51.  One  penny  is  what  part  of  a  pound? 

52.  What  part  of  a  pound  is  2  pence?  o  pence? 
4  pence?  5  pence  ?  6  pence  ?  7  pence  ?  8  pence?  11 
pence?    15  pence?    27  pence  ?    35  pence? 

53.  How  many  pence  are  there  in  1  shilling  and  6 
pence? 

54.  In  2  shillings  and  4  pence,  how  many  pence  ? 

55.  In  4  shillings  and  5  pence,  how  many  pence? 
"   56.  In  5  shillings  and  8  pence,  how  many  pence? 

57.  In  9  shillings  and  11  pence,  how  many  pence  ? 

58.  What  part  of  £1  is  2s.  6d.  ? 

59.  3s.  5d.  is  what  part  of  £1  ? 

Note. — Reduce  the  icJiole  to  pence. 

60.  7s.  8d.  is  what  part  of  £1  ? 

61.  What  is  the  price  of  2  yards  of  cloth,  at  3s. 
4d,  a  yard  ? 

62.  What  will  8  yards  of  cloth  cost,  at  2s.  8d.  a 
yard  ? 

68.  What  will  4  bushels  of  wheat  cost,  at  5s.  9d.  a 
bushel  ?  * 

64.  What  must  you  give  for  4  barrels  of  cider,  at 
2}  dollars    a  barrel  ? 

65.  If  3  bushels  of  wheat  be  divided  between  2 
men,  how  much  would  they  have  apiece  ? 

66.  If  4  bushels  of  corn  be  divided  amSng  5  men, 
how  much  would  they  have  apiece  ? 

67.  If  3  bushels  of  corn  be  divided  among  7  men, 
how  much  would  they  have  apiece  ? 

68.  How  many  nails  are  there  in  1  yard? 

69.  How  many  nails  are  there  in  4  yards  ? 

70.  How  many  nails  are  there  in  5  yards  and  2 
nails  ? 

71.  In  7  yards  and  3  quarters  how  many  quarters? 


120  ARITHMETIC.  [part  i. 

72.  In  4  yardSj  2  quarters,.and  3  nails,  how  many- 
nails  ? 

73.  1  nail  is  what  part  of  a  quarter? 

74.  3  nails  is  what  part  of  a  quarter  ? 

75.  1  nail  is  what  part  of  a  yard  ? 

76.  What  part  of  1  yard  is  3  nails  ?  5  nails  ?  7 
nails  ?    10  nails  ?    15  nails  ? 

77.  In  B  quarters  of  a  yard  how  many  yards  ? 

78.  In  12  quarters  of  a  yard  how  mauy  yards  ? 

79.  In  10  quarters  of  a  yard  how  many  yards? 

80.  In  15  quarters  of  a  yard  how  many  yards? 

81.  In  12  nails  how  many  quarters  of  a  yard  ? 

82.  In  16  nails  how  many  quarters  of  a  yard  ? 
How  many  yards  ? 

83.  In  24  nails  how  many  quarters  of  a  yard? 
How  many  yards  ? 

84.  In  35  quarters  of  a  yard  how  many  yards  ? 
'    85.  In  45  nails  how  many  yards? 

86.  In  63  nails  how  many  yards  ? 

87.  At  2  cents  a  nail  what  would  4  yards  of  cloth 
cost? 

88.  At  2f  dollars  for  1  quarter  of  a  yard,  what 
would  2  yards  cost? 

89.  1  oz.  is  what  part  of  a  lb.  ? 

90.  What  fkrt  of  a  lb.  is  2  oz.?  3  oz.?  4  oz. ?  5 
oz. ?  7  oz.?  10  oz.?  15  oz.? 

91.  What  part  of  a  qr.  of  1  cwt.  is  1  lb.  ?  2  lbs.? 
3  lbs.?  4  1bs..i'  7  lbs.?  9  lbs.?  14  lbs.?  18  lbs.?  23 
lbs.? 

92.  At  3  cents  for  1  oz.  what  would  1  lb.  cost? 

93.  At  3  cents  for  2  oz.  what  would  1  lb.  cost  ? 

94.  At  3  cents  for  8  oz.  what  would  1  lb.  cost? 

95.  At  5  cents  for  10  pz.  what  would  1  lb.  cost? 

96.  At  8  shillings  for  4  lbs.  what  would  10  lbs. 
cost? 

97.  If  a  man  consumes  1  lb.  and  3  oz.  of  meat  in  a 
day,  how  much  would  he  consume  in  a  week? 


PARTI.]  ARITHMETIC.  121 

98.  If  a  man  spends  2|  dollars  in  a  day,  how  much 
would  he  spend  in  a  wcekj? 

99.  If  a  man  travels  3|  miles  in  an  hour,  how  far 
would  he  travel  in  3  hours  ?  How  far  in  7  hours  if 
How  far  in  12  hours? 

100.  If  2  men  start  from  the  same  place,  and  travel 
in  opposite  directions ;  one  at  the  rate  of  o|  miles  in 
an  hour,  and  the  other  4?-  miles;  how  far  will  they  be 
apart  at  the  end  of  1  hour?  How  far  at  the  end  of  2 
hours  ?  How  far  at  the  end  of  3  hours  ?  How  far  at 
the  end  of  7  hours  ? 

101.  Two  men  start  from  the  same  place,  and  travel 
the  same  way;  one  at  the  rate  of  4\  miles  #n  an  hour, 
the  other  at  the  rate  of  4|  miles  in  an  hour;  how  far 
will  they  be  apart  at  the  end  of  1  hour  ?     How  far  in 

2  hours  ?     How  far  in  5  hours?    How  far  in  10  hours? 
How  far  in  o  days,  if  they  travel  10  hours  in  a  day? 

102.  How  many  yards  of  cloth,  at  5  dollars  a  yard, 
must  be  given  for  8  barrels  of  flour,  at  7  dollars  per 
barrel  ? 

103.  What  part  of  a  month  is  1  week  ?     2  weeks  ? 

3  weeks  ? 

104.  What  part  of  a  year  is  1  month?  2  months? 
3  months?  4  months?  5  months?  6  months?  7 
months?  8  months?  9  months?  10  months?  11 
months  ? 

105.  What  part  of  1  month  is  1  day  ?  2  days  ?  3 
days?  7  days?  8  days?  11  days?  15  days?  18 
days  ?    20  days  ?    24  days  ?    27  days  ? 

106.  If  5  bushels  of  oats  will  keep  7  horses  throui!;h 
the  winter,  how  many  bushels  will  it  take  to  keep  12 
horses  the  same  time  ? 

107.  If  you  give  7  men  2i  bushels  of  corn  apiece, 
how  many  bushels  would  it  take  for  the  whole? 

108.  A  man,  failing  in  trade,  was  able  to  pay  his 
creditors  only  4  shillings  on  a  dollar;  how  much  would 
he  pay  on  2  dollars  ?  How  much  on  3  dollars  ?  How 
much  on  7  dollars  ?     How  much  on  10  dollars  ? 


122  ARITHMETIC.  [paet  i. 

109.  A  man,  failing  in  trade,  is  able  to  pay  only  9 
shillings  on  a  pound;  how.  much  would  he  pay  on  a 
debt  of  2  pounds  ?  How  much  on  3  pounds  ?  How 
much  on  12  pounds  ? 

110.  A  man,  failing  in  trade^  is  able  to  pay  only  4 
shillings  and  7  pence  on  a  dollar ;  how  much  would  he 
pay- on  a  debt  of  7  dollars? 

111.  If  6  dollars'  worth  of  provisions  will  serve  3 
men  5  days,  how  many  days  will  it  serve  1  man  ?  How 
many  days  will  it  serve  2  men  ?  How  many  days  will 
it  serve  8  men  ? 

112.  If  10  dollars'  worth  of  provisions  will  serve  7 
men  4  daysk,  how  many  days  will  it  serve  9  men  ? 

113.  If  12  dollars'  worth  of  provisions  will  serve  7 
men  3  days,  how  many  men  would  it  serve  1  day  ? 
How  many  2  days  ?     How  many  8  days  ? 

114.  If  11  dollars'  worth  of  provisions  will  serve  6 
men  8  days,  how  many  men  will  it  serve  5  days  ? 

115.  If  8  dollars'  worth  of  provisions  will  serve  7 
men  5  days,  how  many  days  would  16  dollars'  worth 
of  provisions  last  4  men  ? 

116.  If  1  peck  of  wheat  affords  12  ten-penny  loaves, 
how  many  penny  loaves  may  be  obtained  from  it? 
How  many  two-penny  loaves?  How  many  -three- 
penny loaves  ?     How  many  seven-penny  loaves  ? 

117.  If  1  peck  of  wheat  affords  11  eight-penny 
loaves,  how  many  ten-penny  loaves  will  it  afford  ? 

118.  A  man  having  hired  some  men  and  some  boys, 
agreed  to  give  each  man. 3  shillings,  and  each  boy  2 
shillings;  how  much  would  it  take  to  pay  a  man  and  a 
boy?  How  much  2  men  and  2  boys?  How  much  7 
men  and  7  boys  ? 

119.  A  man  having  18  shillings  to  pay  among  his 
laborers,  would  give  to  every  man  2  shillings,  and  to 
every  boy  1  shilling;  the  number  of  men  and  boys 
was  equal ;  how  many  were  there  of  each  ? 

120.  A  gentleman  having  50  shillings  to  pay  among 
his  laborers,  would  give  to  every  man  8  pence,  and  to 


PART  I.]  ARITHMETIC.  123 

every  boy  4  pence;  the  number  of  men  and  boys  Was 
equal;  how  mftny  were  there  of  each  ? 

121.  Two  men  bought  a  bushel  of  corn,  one  gave 
1  shilling,  the  other  2  shillings ;  what  part  of  ^e 
whole  did  each  pay  ?  What  part  of  the  corn  must 
each  have  ? 

122.  Two  men  1>ought  a  barrel  of  flour  for  8  dollars ; 
one  gave  3  dollars,  the  other  5  dollars ;  what  part  (^id 
^ach  pay  ?  and  what  part  must  each  have  ? 

128.  Three  men,  A,  B,  and  C,  rented  a  garden  ;**A 
paid  6  dollars,  B  5  dollars,  and  G  9  dollars  ;  how  much 
did  they  all  pay?  What  part  of  the  whole  did  each 
pay?  They  sold  the  produce  for  40  dollars;  what 
part  of  it  must  each  have  ?  What  did  each  one's 
share  amount  to  ? 

124.  Three  men  bought  a  lottery  ticket  for  10  dol- 
lars ;  the  first  gave  3  dollars,  the  second  5  dollars,  and 
the  third  2  dollars;  they  drew  a  prize  of  120  dollars; 
what  was  each  man's  share  ? 

125.  Two  men  rented  a  pasture  for  58  dollars;  one 
put  in  7  horses,  and  the  other  3  horses;  what  ought 
each  to  pay  ? 

126.  Three  men  commenced  trade  together;  they 
put  in  money  in  the  following  proportion ;  the  first  3 
dollars,  as  often  as  the  second  put  in  4,  and  as  often  as 
the  third  put  in  5;  they  gained  87  dollars;  what  was 
each  man's  share  of  the  gain?  * 

127.  Two  men  rented  a  pasture  for  32  dollars ;  the 
first  put  in  3  sheep  for  4  months,  the  second  put  in  4 
sheep  for  5  months ;  how  much  ought  each  to  pay  ? 

Note. — Three  sheep  for  4  months  is  the  same  as  12 
sheep  for  1  month,  4  sheep  for  5  months  is  the  same 
as  20  sheep  for  1  month.  This  question  is,  therefore, 
the  same  as  if  1  man  put  in  12  sheep,  and  the  other 
20  sheep. 

128.  Two  men,  A  and  B,  traded  in  company :  A 
put  in  1  dollar  for  4   months,  and  B   2   dollars  for  3 


124  ARITHMETIC.  [part  i. 

months,    and    they    gained   ninety  cents ;   how  many 
cents  must  each  have  ? 

129.  Three  men,  A,  B,  and  C,  traded  in  company; 
and  put  in  money  in  the  following  proportions  .  A  put 
in  4  dollars  as  often  as  B  put  in  3,  and  as  often  as  C 
put  in  2 ;  A's  money  was  in  2  months,  B's  3  months, 
and  C's  4  months,  and  they  gained  100  dollars ;  what 
was  each  one's  share  ? 

130.  Two  men,  A  and  B,  traded  in  company;  A 
put  in  2  dollars  as  often  as  B  put  in  3 ;  A's  money  was 
employed  7  months,  and  B's  money  5  months ;  they 
gained  58  dollars;  what  was  each  man's  share  of  the 
gain  ? 

131.  Three  men,  A,  B,  and  C,  traded  in  company, 
and  put  in  money  in  the  following  proportions  :  A  put 
in  2  dollars  as  often  .as  B  put  in  4,  and  as  often  as  C 
put  in  6.  B's  money  was  in  twice  as  long  as  C's,  and 
A's  two  times  as  long  as  B's;  they  gained  88  dollars; 
what  was  each  one's  share  of  the  gain  ? 

Note. — Interest  is  a  reward  or  premium  allowed  by  a 
debtor  to  a  creditor  for  the  use  of  money.  The  interest  for 
1  year,  as  established  by  law  in  some  of  the  States,  is  6 
cents  on  a  dollar,  6  dollars  on  a  hundred  dollars,  or  in  fine 
■j§iy  of  the  sum  whatever  be  the  denomination.  It  is  called 
6  per  cent.,  that  is,  6  on  the  hundred,  because  it  is  always 
reckoned  by  the  hundred.  So  3  per  cent.,  4  per  cent.,  etc., 
signify  j§^,  y^^^,  etc.,  or  so  much  on  the  hundred.  The 
teacher  can  vary  the  examples  to  illustrate  the  different 
rates  per  cent. 

132.  The  interest  of  1  dollar  being  6  cents  for  1 
year,  what  is  the  interest  of  7  dollars  for  the  same 
time?  What  is  the  interest  of  10  dollars?  of  15  dol- 
lars? of  20  dollars?  of  30  dollars?  of  50  dollars? 
of  75  dollars  ?    of  100  dollars  ?    of  118  dollars  ? 

133.  If  the  interest  of  1  dollar  is  6  cents  for  1  year, 
what  would  it  be  for  2  years  ?  What  would  be  the 
interest  of  8  dollars  for  2  years  ?  of  17  dollars  ?  of 
43  dollars  ? 


PART  I.]  ARITHMETIC.  125 

134.  If  the  interest  of  100  dollars  is  6  dollars  for  a 
year,  what  would  be  the  interest  of  50  dollars  for  the 
same  time  ?  of  2  hundred  ?  of  3  hundred  ?  of  4  hun- 
dred ?  of  1  hundred  and  50  ?  of  2  hundred  and  50  ? 

135.  If  the  interest  of  100  dollars  is  6  dollars  for  1 
year,  what  would  be  the  interest  of  it  for  G  mouths  ? 
for  3  months?  for  4  months?  for  8  months?  for  9 
months?  for  1  month?  for  2  months?  for  5  months? 
for  7  months?    for  10  months  ?    for  11  months? 

136.  What  is  the- interest  of*  100  and  32  dollars  for 
2  years,  at  6  per  cent.? 

137.  What  is  the  interest  of  300  dollars  for  1  year 
and  6  months,  at  6  per  cent.  ? 

138.  What  is  the  interest  of  1  dollar  for  6  months, 
at  6  per  cent.  ?  what  for  2  months  ?  what  for  1  month  ? 
what  for  3  months  ?  4  months  ?  5  months  ?  7  months? 
9  months?  11  months? 

139.  What  is  the 'interest  of  57  dollars  for  1  year 
and  7  months,  at  6  per  cent.  ? 

140.  What  is  the  interest  of  200  and  67  dollars  for 

1  year  and  4  months,  at  6  per  cent.  ? 

141.  If  the  interest  of  1  year  is  6  per  cent.,  what 
would  be  the  per  cent  for  2  years  ?  for  3  years  ?  for  6 
months  ?  for  2  months  ?  for  1  month  ?  for  4  months  ? 
for  5  months  ?  for  7  months  ?  for  8  months  ?  for  9 
fiionths  ? 

142.  If  the  interest  of  2  months,  or  60  days,  is  1  per 
cent,,  what  would  be  the  per  cent  for  20  days?  what 
for  40  days  ?  what  for  15  days  ?  what  for  45  days  ? 
what  for  12  days?  what  for  10  days?  what  for  5 
days  ? 

143.  What  is  the  interest  of  100  and  37  dollars  for 

2  years  3  months  and  20  days  ? 

144.  A  can  do  a  piece  of  work  in  2  days ;  how 
much  of  it  can  he  do  in  1  day  ? 

145.  B  can  do  a  piece  of  work  in  4  days;  how 
much  of  it  can  he  do  in  1  day  ? 

146.  If  A  can  do  J  of  a  piece  of  work  in  1  day, 


126  ARITHMETIC.  [parti. 

and  B  can  do  i  of  it  in  1  day,  how  much  would  both 
do  in  a  day  ?  How  long  would  it  take  them  both 
together  to  do  the  whole  ? 

147.  If  1  man  can  do  a  piece  of  work  in  2  days, 
and  another  in  3  days,  how  much  of  it  would  each  do 
in  a  day?  How  much  would  both  together  do?  How 
long  would  it  take  them  both  to  do  the  whole  ? 

148.  A  cistern  has  2  cocks;  the  first  will  fill  it  in  3 
hours,  the  second  in  6  hours ;  how  much  of  it  would 
each  fill  in  an  hour  ?  How  much  would  both  together 
fill?     How  long  would  it  take  them  both  to  fill  it  ? 

149.  A  man  and  his  wife  found  by  experience,  that, 
when  they  were  both  together,  a  bushel  of  meal  would 
last  them  only  2  weeks ;  but  when  the  man  was  gone, 
it  would  last  his  wife  5  weeks;  how  much  of  it  did  both 
together  consume  in  1  we<ek  ?  What  part  did  the  woman 
alone  consume  in  1  week  ?  What  part  did  the  man 
alone  consume  in  1  week  ?  How*long  would  it  last  the 
man  alone? 

150.  If  1  man  could  build  a  piece  of  wall  in  5  days, 
and  another  man  could  do  it  in  7  days,  how  much  of 
it  would  each  do  in  1  day  ?  How  many  days  would  it 
take  them  both  to  do  it  ? 

151.  A  cistern  has  3  cocks;  the  first  would  fill  it  in 
3  hours;  the  second  in  6  hours;  the  third  in  4  hours ; 
whrft  part  of  the  whole  would  each  fill  in  1  hour  ?  and 
how  long  would  it  take  them  all  to  fill  it,  if  they  were 
all  running  at  once? 

152.  A  and  B  together  can  build  a  boat  in  8  days, 
and  with  the  assistance  of  C  they  can  do  it  in  5  days ; 
how  much  of  it  can  A  and  B  build  in  1  day  ?  How 
much  of  it  can  A,  B,  and  C,  build  in  1  day  ?  How 
much  of  it  can  C  build  alone  in  1  day?  How  long 
would  it  take  C  to  build  it  alone  ? 

153.  Suppose  I  would  line  8  yards  of  broadcloth  that 
is  1^  yards  wide,  with  shalloon  that  is  f  of  a  yard  wide; 
how  many  yards  of  the  shalloon  will  line  1  yard  of  the 
broadcloth  ?     How  many  yards  will  line  the  whole  ? 


PAKT  I.]  ARITHMETIC.  127 

154.  If  7  yards  of  cloth  cost  13  dollars,  what  will 
10  yards  cost  ? 

155.  If  the  wages  of  25  weeks  come  to  75  dollars, 
what  will  be  the  wa<res  of  7  weeks? 

156.  If  S.tons  of  hay  will  keep  7  horses  three  months, 
how  much  will  keep  12  horses  the  same  time  ? 

157.  If  a  staff  4  feet  long  casts  a  shadow  6  feet  long, 
what  is  the  length  of  a  pole  that  casts  a  shadow  58  feet 
at  the  same  time  of  day  ? 

158.  If  a  stick  8  feet  long  casts  a  shadow  2  feet  in 
length,  what  is  the  height  of  a  tree  which  casts  a  shadow 
42  feet  at  the  same  time  of  day  ? 

159.  x\t  six  dollars  per  week,  how  many  months 
board  can  I  have  for  100  dollars  ? 

160.  A  ship  has  sailed  24  miles  in  4  hours;  how 
long  will  it  take  her  to  sail  150  miles  at  the  same  rate? 

161.  30  men  can  perform  a  piece  of  work  in  20  days  ; 
how  many  men  will  it  take  to  perform  the  same  work 
in  8  days '/ 

162.  17  men  can  perform  a  piece  of  work  in  25 
days ;  in  how  many  days  would  5  men  perform  the  same 
work  ? 

163.  A  hare  has  76  rods  the  start  of  a  greyhound, 
but  the  greyhound  runs  15  rods  to  10  of  the  hare;  how 
many  rods  must  the  greyhound  run  to  overtake  the 
hare  ? 

164.  A  garrison  has  provision  for  8  months,  at  the 
rate  of  15  ounces  per  day;  how  much  must  be  allowed 
per  day,  in  order  that  the  provision  may  last  11  months? 

165.  If  8  men  can  build  a  wall  15  rods  in  length  in 
10  days,  how  many  men  will  it  take  to  build  a  wall  45 
rods  in  length  in  5  days  ? 

166.  If  a  quarter  of  wheat  affords  60  ten-penny 
loaves,  how  many  eight-penny  loaves  may  be  obtained 
from  it  ? 

167.  Said  Harry  to  Dick,  My  p\irse  and  money 
together  are  worth  16  dollars,  but  the  money  is  worth  7 


128  ARITHMETIC.  [part  i. 

times  as  much  as  the  purse;  how  much  money  was  there 
in  the  purse  ?  and  what  is  the  value  of  the  purse  ?     , 

168.  A  man  being  asked  the  price  of  his  horse, 
answered,  that  his  horse  andsaddle  together  were  worth 
100  dollars,  but  the  horse  was  worth  9  times  as  much 
as  the  saddle;  what  was  each  worth  ? 

169.  A  man  having  a  horse,  a  cow,  and  a  sheep,  was 
asked  what  was  the  value  of  each.  He  answered,  that 
the  cow  was  worth  twice  as  much  as  the  sheep,  and  the 
horse  8  times  as  much  as  the  sheep,  and  that  all  together 
were  worth  60  dollars ;  what  was  the  value  of  each  ? 

170.  A  man  bought  an  apple,  an  orange  and  a  melon, 
for  21  cents ;  for  the  orange  he  gave  twice  as  much  as 
for  the  apple,  and  for  the  melon  he  gave  twice  as  much 
as  for  the  orange;  how  much  did  he  give  for  each  ? 

171.  If  80  dollars'  worth  of  provisions  will  serve  20 
men  24  days,  how  many  days  will  100  dollars'  worth 
of  provisions  serve  80  men  1 

172.  There  is  a  pole  \  and  \  under  water,  and  10 
feet  out ;  how  long  is  the  pole  ? 

173.  In  an  orchard  of  fruit  trees,  \  of  them  bear 
apples,  \  of  them  bear  plums,  \  of  them  pears,  7  of 
them  peaches,  and  3  of  them  cherries  ;  how  many  trees 
are  there  in  the  whole,  and  how  many  of  each  sort? 

174.  A  farmer  being  asked  how  many  sheep  he  had, 
answered,  that  he  had  them  in  4  pastures  ;  in  the  first 
he  had  \  of  his  flock  ;  in  the  second  i;  in  the  third  i; 
and  in  the  fourth  15;  how  many  sheep  had  he  ? 

175.  A  man  driving  his  geese  to  market,  was  met  by 
another,  who  said.  Good-morrow,  master,  with  your 
hundred  geese.  Says  he,  I  have  not  a  hundred;  but  if 
I  had  half  as  many  more  as  I  now  have,  and  two  geese 
and  a  half,  I  should  have  a  hundred;  how  many  had 
he? 

176.  What  number  is  that,  to  which  if  its  half  be 
added  the  sum  will  be  60  ? 

177;  What  number  is  that,  to  which  if  its  third  be 
added  the  sum  will  be  48  ? 


PART  I.]  ARITHMETIC.  129 

178.  What  number  is  that,  to  which  if  its  fifth  be 
added  the  sum  will  be  54  ? 

179.  What  number  is  that,  to  which  ifits  half  and  its 
third  be  added  the  sum  will  be  55  ? 

^  180.  A  man  being  asked  his  age,  answered,  that  if 
Its  half  and  its  third  were  added  to  it,  the  sum  would  be 
77  ;  what  was  his  age  ? 

181.  What  Dumber  is  that  which  being  increased  by 
Its  half,  its  fourth,  and  eighteen  more,  will  be  doubled? 

182.  A  boy  being  asked  his  age,  answered,  that  if 
^  and  i  of  his  age,  and  20  more  were  added  to  his 
age,  the  sum  wo«ld  be  3  times  his  age.     What  was  his 


age  / 


18o.  A  man  being  asked  how  many  sheep  he  had, 
answered,  that  if  he  had  as  many  more,  ^  as  many  more, 
and  2  J  sheep,  he  should  have  100.     How  many  had  he*'' 


ARITHMETIC. 


PART    II. 

KEY. 

The  Key  contains  remarks  on  the  principles  employed, 
and  illustrations  of  the  manner  of  solving  the  examples  in 
each  section.  All  the  most  difficult  of  the  practical  exam- 
ples are  solved  in  such  a  manner  as  to  show  the  principles 
by  which  they  are  performed.  Care  has  been  taken  to 
select  examples  for  solution,  tliat  will  explain  those  which 
are  not  solved..  Many  remarks  with  refjard  to  the  manner 
of  illustrating  the  })rinciples  to  the  pupils,  are  inserted  in 
their  proper  places,      , 

Instructors  who  may  never  have  attended  to  fractions, 
need  not  be  afraid  to  undertake  to  teach  this "  book.  The 
author  flatters  hihiself  that  the  principles  are  so  illustrated, 
and  the  processes  are  made  so  simple,  that  any  one,  who 
shall  undertake  to  teach  it,  will  find  himself  familiar  "with 
fractions  before  he  is  aware  of  it,  although  he  knew  nothing 
of  them  before ;  and  that  every  one  will  acquire  a  facility 
in  solving  questions,  which  he  never  before  possessed. 

The  reasoning  used  in  performing  these  small  examples, 
is  precisely  the  same  as  that  used  upon  large,  ones.  And 
when  any  one  finds  a  difficulty  in  solving  a  question,  he 
will  remove  it  much  sooner,  and  much  more  effectually, 
by  taking  a  very  small  example  of  the  same  kind,  and 
observing  how  he  does  it,  than  by  recurring  to  a  rule. 

The  practical  examples  at  the  commencement  of  each 
section  and  article,  are  generally  such  as  to  show  the  pupil 
w^hat  the  com])ination  is,  and  how  he  is  to  perform  it. 
This  will  teach  the  pupil  gradually  to  reason  upon  abstract 
numbers.  In  each  combination,  there  are  a  few  abstract 
examples  without  practical  ones,  to  exercise  the  learnei*  in 
the  combinations,  after  he  knows  what  these  combinations 
are.     It  would  be  an  excellent  exercise  for  the  pupil  to  put 

(131J 


132  KEY.  [PAET II. 

these  into  a  practical  form  when  he  is  reciting.  For  instance, 
■when  the  question  is,  how  many  are  5  and  3  ?  Let  him 
make  a  question  in  this  way :  if  an  orange  cost  five  cents, 
and  an  apple  3  cents,  what  would  they  both  come  to  ?  This 
may  be  done  in  all  cases. 

The  examples  are  often  so  arranged,  that  several  depend 
on  each  other,  so  that  the  preceding  explains  the  following 
one.  Sometimes  also,  in  the  same  example,  there  are  sev- 
eral questions  asked,  so  as  to  lead  the  pupil  gradually  from 
the  simple  to  the  more  difficult.  It  would  be  well  for  the 
pupil  to  acquire  the  habit  of  doing  this  for  himself,  when 
difficult  questions  occur. 

The  operations  can  be  illustrated  by  counters,  or  marks 
on  the  blackboard,  according  to  the  necessity  of  the  pupils. 
These  illustrations  will  be  less  necessary  as  the  pupils 
advance  in  the  work;  but  a  frequent  reference  to  them 
throughout  most  of  the  book  will  be  useful,  in  fixing  more 
clearly  in  mind  the  principles  involved  in  the  operations. 

The  book  may  be  used  in  classes  where  it  is  convenient. 
The  pupil  may  answer  the  questions  with  the  book  before 
him  or  not,  as  the  instructor  thinks  proper.  A  very  useful 
mode  of  recitation  is  for  the  instructor  to  read  the  example 
to  the  whole  class,  and  then,  allowing  sufficient  time  for 
them  to  perform  the  question,  call  upon  some  one  to  answer 
it.  In  this  manner,  all  the  pupils  will  be  obliged  to  perform 
the  example,  because  they  do  not  know  who  is  to  answer  it. 
In  this  way  it  will  be  best  for  them  to  answer  without  the 
book. 

It  will  often  be  well  to  let  the  elder  pupils  hear  the 
younger.  This  will  be  a  useful  exercise  for  them,  and  an 
assistance  to  the  instructor. 


SECTION  I. 

A.  This  section  contains  addition  and  subtraction.  The 
first  example  may  be  solved  by  means  of  beans,  peas,  etc., 
or  by  means  of  the  blackboard.  The  former  method  is 
preferable,  if  the  pupil  be  very  young,  not  only  by  the 
examples  in  the  first  part  of  this  section,  but  by  the  first 
examples  in  all  the  sections. 

The  pupil  will  probably  solve  the  first  examples  without 
any  instruction. 


SEC.  II.]  KEY.  133 

B  &  C.  The  articles  B  and  C  contain  the  common  addi- 
tion table  as  far  as  the  first  10  numbers.  In  the  first,  the 
numbers  are  placed  in  order;  and  in  the  second,  out  of 
order. 

The  pupil  should  study  these  until  he  can  find  the 
answers  readily,  and  then  he  should  commit  the  answers 
to  memory.  > 

D.  In  this  article  the  numV)er8  are  laro;er  than  in  the 
preceding;  and,  in  some  instances,  three  or  more  numbers 
are  added  togetlier.  In  the  abstract  examples,  the  numbers 
from  one  to  ten  are  to  be  added  to  the  numbers  from  ten  to 
twenty. 

E.  This  article  contains  subtraction. 

F.  This  article  is  intended  to  make  the  pupil  familiar 
with  adding  the  first  nine  numbers  to  all  others.  The  pupil 
should  study  it  until  he  can  answer  the  questions  very 
readily. 

G.  In  this  article  all  the  preceding  arc  combined  together, 
and  the  numbers  from  1  to  10*  are  added  to  all  numbers 
from  20  to  100,  and  subtracted  in  the  same  manner. 

18.  57  and  G  are  G3,  and  3  are  66,  and  5-  are  71,  and  2 
are  73,  less  8  are  05. 

H.  This  article  contains  practical  questions  which  show 
the  application  of  all  the  preceding  articles. 

6.  37  less  5  are  32,  less  8  are  24,  less  6  (which  he  kept 
hiinself )  are  18;  consequently  he  gave  18  to  the  third  boy. 

SECTION  II. 

This  section  contains  multiplication.  The  pupil  will  see 
no  difi'erence  between  this  and  addition.  It  is  best  that  he 
should  not  at  first,  though  it  may  be  well  to  explain  it  to 
him  after  a  while. 

A.  This  article  contains  practical  questions,  which  the 

pupil  will  readily  answer. 

*  Figures  are  used  in  the  key.  because  the  instructor  is  supposed 
to  be  acquainted  with  tliem.  They  are  not  used  in  the  first  part  of  the 
book,  because  the  pupil  would  not  understand  them  so  well  as  he  will 
the  words. 


134  KEY.  -      [part  n. 

I.  Three  yards  will  cost  3  times  as  much  as  1  yard. 

N.  B. — Be  careful  to  make  the  pupil  give  a  similar  reason 
for  multiplication,  both  in  this  article  and  elsewhere. 

II.  A  man  will  travel  4  times  as  far  in  4  hours  as  he  will 
in  1  hour. 

15.  There  are  4  times  as  many  feet  in  4  yards  as  in  1 
yard,  or  4  times  3-  feet. 

B.  This  article  contains  the  common  multiplication  table, 
as  far  as  the  product  of  the  first  ten  numbers.  The  pupil 
should  find  the  answers  once,  or  twice  through,  until  he 
can  find  them  readily,  and  then  let  him  commit  them  to 
memory. 

C.  This  article  is  the  same  as  the  preceding,  except  in 
this,  the  numbers  are  out  of  their  naturaLprder. 

D.  In  this  article,  multiplication  is  applied  to  practical 
examples.  They  are  of  the  same  kind  as  those  in  article  A 
of  this  section. 

12.  There  are  8  times  as  many  squares  in  8  rows,  as  in  1 
row.     8  times  8  are  64.  •  ^ 

13.  There  are  6  times  as  many  farthings  in  6  pence,  as 
in  1  penny.     6  times  4  are  24. 

17.  12  times  4  are  48. 

23.  There  are  3  times  as  many  pints  in  3  quarts  as  in  1 
quart,  3  times  2  are  6.  And  in  6  pints  there  are  6  times 
4  gills,  or  24  gills. 

28.  In  3  gallons  there  are  12  quarts,  and  in  12  quarts 
there  are  24  pints. 

31.  In  2  gallons  are  8  quarts  ;  in  8  quarts  16  pints ;  in  16 
pints  64  gills.     16  times  4  are  64. 

35.  In  1  gallon  are  32  gills ;  and  32  times  2  cents  are  64 
cents.  Or,  1  pint  will  cost  8  cents,  and  there  are  8  pints 
in  a  gallon.     8  times  8  are  64. 

38.  They  will  be  2  miles  apart  in  1  hour,  4  miles  in  2 
hours,  etc. 


SECTION  III. 

A.  This  section  contains  division.  The  pupil  will 
scarcely  distinguish  it  from  multiplication.  It  is  not  im- 
portant that  he  should  at  first. 


SEC.  III.]  KEY.  135 

The  pupil  will  be  able  to  answer  these  questions  by  the 
multiplication  table,  if  he  has  committed  it  to  memory 
thoroughly. 

B.  In  this  article  the  pupil  obtains  the  first  ideas  of 
fractions,  and  learns  the  most  important  of  the  terms  which 
arc  applied  to  fractions.*  The  pupil  has  already  been 
accustomed  to  look  upon  a  collection  of  units,  as  formiufj;  a 
number,  or  as  being  itself  a  part  of  another  number.  He 
knows,  therefore,  that  one  is  a  part  of  every  number,  and 
that  ever}^  number  is  a  part  of  every  numljor  larger  than 
itself  As  every  number  may  have  Ji  variety  of  parts,  it  is 
necessary  to  give  names  to  the  different  parts,  in  order  to 
distinguish  tliem  from  each  other.  The  parts  receive  their 
names,  according  to  the  numl»er  of  parts  into  which  any 
numljcr  is  divided.  If  the  number  is  divided  into  two'  equal 
parts,  the  parts  arc  called  halves  ;  if  it  is  divided  into  three 
equal  parts,  they  are  called  thirds ;  if  into  four  parts, 
fourths,  etc. ;  and  having  divided  a  number  into  parts,  we 
can  take  as  many  of  the  parts  as  we  choose.  If  a  number 
is  divided  into  five  equal  parts,  and  three  of  the  parts  are 
taken,  the  fraction  is  called  three  Jifths  of  the  number. 
The  name  shows  at  once  into  how  many  parts  the  number 
is  to  be  divided,  and  how  many  parts  are  taken. 

The  examples  in  this  book  are  so  arranged  that  the 
names  will  usuall}^  show  the  pupil  how  the  operation  is  to 
be  performed.  In  this  section,  although  the  pupil  is  taught 
to  divide  numbers  into  vjlrious  parts,  he  is  not  taught  to 
notice  any  fractions,  except  those  where  the  numbers  are 
divided  into  their  simple  units,  which  is  the  most  simple 
kind. 

It  will  be  best  to  use  beans,  pebbles,  etc.,  first. 

20.  Since  1  is  1  third  of  3,  2  must  be  2  thirds  of  3. 

34.  Illustrate  by  grouping  the  marks  or  counters  by 
threes. 

Pi'oceed  in  the  same  manner  with  the  other  divisions. 

This  being  one  of  the  most  useful  combinations,  and  one 
but  very  little  understood  by  most  people,  especially  when 
applied  to  large  numbers*,  the  pupil  must  be  made  perfectly 
familiar  with  it.  Ask  qjaestions  like  those  in  the  book  for 
large  numbers,  and-  also  some  like  the  following:  What 
part  of  7  is  18?  the  answer  will  be  V. 

*  As  soon  as  the  terms  applied  to  fractions  are  fully  comppohended, 
the  operations  on  them  are  as  simple  as  those  on  whole  numbers. 


136  KEY.  [part  ii. 

C.  The  first  ten  figures  are  here  explained.  They  are 
used  as  an  abridged  method  of  writing  numbers,  and  not 
with  any  reference  to  their  use  in  calculating. 

This  article  is  only  a  continuation  of  the  last.  All  the 
numbers  from  1  to  100  are  introduced  into  the  two  articles, 
and  are  divided  by  all  the  numbers  from  1  to  10 ;  except 
that  some  of  the  largest  are  not  divided  by  some  of  the 
smallest. 

2.  The  pupil  answers  first,  how  many  times  2  is  con- 
tained in  12,  then  how  many  times  3. 

D.  These  examples,  which  are  similar  to  those  in  article 
A  of  this  section,  are  solved  in  the  same  manner. 

5.  It  would  take  as  many  hours  as  3  miles  are  contained 
in  10  miles.     3  hours  and  I  of  an  hour. 

20.  They  cost  as  many  cents  as  there  are  3  apples  in  30 
apples ;  that  is,  10  cents. 

21.  12  dollars  a  month  ;  and  12  dollars  a  month  is  3  dol- 
lars a  week ;  that  is,  18  shillings  a  week,  which  is  3  shil- 
lings a  day. 

26.  The  whole  loss  was  35  dollars,  which  was  7  dollars 
apiece. 


SECTION  IV. 

A.  This  article  contains  multiplication  simply.  It  is 
repeating  a  number  a  certain  number  of  times  and  a  part 
of  another  time. 

14.  6  times  5  are  30,  and  |  of  5  are  3,  which  added  to 
30  make  33. 

B.  In  this  article  the  pupil  is  taught  to  change  a  certain 
number  of  twos  into  threes,  threes  into  fives,  etc.  This 
article  combines  all  the  preceding  operations. 

24.  4  cords  of  wood  will  cost  28  dollars,  and  f  of  a  cord 
will  cost  2  dollars,  which  makes  30  dollars.  30  dollars  will 
buy  3  hundred  weight  of  sugar  and  f  of  another  hundred 
weight. 

29.  7  times  8  are  56,  and  f  of  8  are  5,  which  added  to 
56  make  61 ;  61  are  6  times  9,  and  ^  of  9. 

C.  1.  4  bushels  of  apples,  at  3  shillings  a  bushel,  come 
to  12  shillings  ;  and  12  shillings  are  2  dollars. 


SEC.  v.]  KEY.  137 

2.  The  2  lemons  come  to  8  cents,  and  8  cents  will  buy  4 
apples,  at  2  cents  apiece. 

This  is  usually  called  Barter.  The  {j^eneral  principle  is 
to  find  what  the  article  will  come  to,  whose  price  and  quan- 
tity are  given,  and  then  to  find  how  much  of  the  other 
arlicle  that  money  will  buy. 

G.  If  2  npples  cost  4  cents,  1  will  cost  2  cents,  and  4  will 
cost  8  cents.  Or  4  apples  will  cost  2  times  as  much  as  2 
apples. 

22.  Find  how  many  times  2  pears  arc  contained  in  20 
poars,  which  is  10  times.  10  times  3  cents  are  30  cents. 
Or  first  find  what  20  pears  would  come  to  at  3  cents  apiece ; 
and  since  it  is  2  for  3  cents,  instead  of  1  for  3  cents,  the 
price  will  be  half  as  much. 

23.  See  how  many  times  you.  can  have  5  cents  in  30 
cents,  and  you  can  buy  so  many  times  3  eggs.  30  is  6 
times  5,  and  6  times  3  are  18.     18  eggs. 

24.  10  dollars  a  week,  and  40  dollars  a  month. 

25.  5  dollars  are  30  shillings,  which  is  10  shillings  a  day. 
6  shillings  is" equal  to  1  dollar  in  some  of  the  States. 

26.  5  dollars  apiece. 


SECTION  V. 

In  this  section  the  principle  effractions  is  applied  to 
larger  numbers,  btrt  such  as  are  divisible,  into  the  parts 
proposed  to  be  taken.  The  pupil,  who  is  familiar  with 
what  precedes,  will  easily  understand  the  examples  in  this 
section.  They  require  nothing  but  dinsion  and  multipli- 
cation. 

A.  Let  the  pupil  explain  each  example  in  the  following 
manner:  What  is  1  sixth  of  18?  Aus.  3.  Why?  Be- 
cause 6  times  3  are  18;  therefore  if  you  divide  18  into  6 
equal  parts,  one  of  the  parts  will  be  3. 

The  pupil  will  be  very  likely  to  say  3  is  the  6th  part  of 
18,  because  3  times  6  arc  18.  Be  careful  to  make  him  say 
it  the  other  way,  viz.,  6  times  3  are  18. 

14.  1  third  of  9  is  3  ;  |  is  2  times  as  much  as  ^,  there- 
fore I  of  9  is  6. 

19.  1  barrel  will  cost  {  part  of  12  dollars;  3  barrels  will 
cost  f  of  12  dollars ;  7  barrels  will  cost  |  of  12  dollars. 

37.  What  is  f  of  32?  ^  of  32  is  4,  f  are  5  times  4,  or  20. 


138  KEY.  [part  ii. 

B.  11.  I  of  20  is  4;  I  are  7  times  4,  of  28;  and  28  is  4 
times  6  and  f  of  6. 

C.  3.  1  half  of  10  is  5;  f  of  10  are  4;  5  and  4  are  9.  He 
gave  away  9,  and  had  1  left. 

4.  1  yard  will  cost  ^,of  what  3  yards  cost.  ^  of  6  dol- 
lars is  2  dollars. 

5.  2  yards  will  cost  1  half  of  what  4  cost;  or  6  dollars. 

6.  3  apples  will  cost  ^  of  what  9  cost,  or  6  cents. 

7.  2  is  f  of  3 ;  therefore  2  oranges  will  cost  |  of  what  3 
cost,     f  of  18  cents  are  12  cents. 

■    8.  f  of  25"  are  20.     The  10  apples  cost  20  cents,  which 
was  2  cents  apiece. 

11.  f  of  42  are  12,  and  6  times  12  are  72.     72  dollars. 

13.  3  is  f  of  4;  f  of  12  dollars  are  9  dollars;  or  4  yards 
at  12  dollars  is  3  dollars  a  yard,  and  9  dollars  for  3  yards. 

14.  Solved  like  the  13.     Ans.  15  cents. 

15.  Since  1  is  ^  of  3,  7  is  ^  of  3:  |  of  15^  cents  are  35 
cents ;  or  3  oranges  at  15  cents,  is  5  cents  apiece.  7  times 
5  cents  are  35  cents. 

Note. — In  questions  of  this  kind,  it  is  generally  the  sim- 
plest way  to  find  what  1  article  will  cost,  then  it  may  easily 
"be  told  how  much  any  number  will  cost. 

19.  4  men  Avould  do  it  in  1  half  the  time  that  2  would  do  it. 
Or,  you  may  say,  if  2  men  would  do  it  in  6  days,  1  man  would 
do  it  12  days,  and  4  men  in  I  of  that  time,  or  3  days. 


SECTION  VI. 

A.  4.  2  halves  of  any  number  ;nake  the  whole  number. 
Therefore  2  is  1  half  of  2  times  2 ;  or  4.  It  is  i  of  4  times 
2,  or  8. 

Let  the  pupil  answer  these  questions  in  the  following 
manner:  4  is  }  of  3  times  4;  3  times  4  are  12.  5  is  if  of 
7  times  5  ;  7  times  5  are  35. 

B.  2.  4  is  2  times2. 
4.  6  is  2  times  3. 

16.  2  thirds  of  any  number  is  twice  as  much  as  ^  of  the 
same  number.  If  4  is  f  of  some  number,  then  1  half  of  4 
or  2  is  i  of  that  number ;  2  is  J  of  6 ;  therefore  4  is  f  of  6. 


SEC.  VI.]  KEY.  139 

20.  If  6  is  f  of  a  number,  |^  of  6  or  2  is  J  of  the  same 
number ;  2  is  ^  of  8  ;  therefore  6  is  f  of  8. 

23.  It  is  evident  that  ^  of  a  pound  will  cost  only  J  of 
what  ^  will  cost.  If  ^  co.st  6  cents,  ^  will  cost  2  cents, 
and  the  whole  pound  14  cents. 

26*.  It  will  probably  be  perceived  by  this  time,  that  f  of 
a  number  being  given,  it  is  necessary  to  find  \,  and  then 
the  number  is  easily  found ;  4  being  f ,  2  is  i,,  and  2  is  | 
of  14. 

45.  24  being  |,  i  of  24  or  3  will  be  ^  ;  3  is  ^  of  27. 

C.  6.  20  being  f ,  5  is  j,  and  5  is  4  of  35  ;  and  35  is  5 
times  6,  and  f  of  6. 

D.  4.  18  is  three  times  6,  and  6  is  1  of  4  times  6,  or  24. 
A71S.  24  dollars. 

6.  54  is  I  of  48  ;  12  yards  at  48  dollars  i^  4  dollars  a  yard. 
He  cainod  6  dollars. 

7."  10  feet  is  |  of  15  feet. 

8..  If  I  are  under  water,  there  must  be  ^  out  of  the  water. 
4  is  i  of  12. 

9.  Iff  arc  underwater,  there  must  be  f  out  of  the  water. 
6  is  I  of  10. 

10.  f  and  f  ai'c  |.  f  bear  cherries  and  peaches ;  conse- 
quently, the  10  which  bear  plums  must  be  the  other  f .  10 
is  f  of  35.     10  bear  peaches,  and  15  bear  cherries. 

11.  |.  and  |,  and  |,  and  ^,  are  | ;  therefore  12  must  be 
the  other  f  of  the  whole.     The  whole  number  is  54. 


MISCELLANEOUS   EXAMPLES. 

6.  The  greyhound  gains  upon  the  fox  4  rods  in  a  minute. 
It  will  take  him  20  minutes  to  gain  80  rods. 

8.  I  of  24.    Or  you  may  say,  1  sheep  would  cost  3  dollars, 
and  3  sheep  9  dollars. 

9.  30  horses  will  eat  10  times  as  much  as  3  horses. 

11.  10  dollars  apiece,  and  2  dollars  a  yard. 

12.  5  dollars  for  1  week,  20  dollars^  for  a  month,  and  25 
dollars  for  5  weeks. 

14.  It  would  take  them  5  times  as  long  to  eat  40  bushels, 
as  it  would  to  eat  8  bushels. 

15.  4  horses  would  eat  4  bushels  in  3  days,  and  it  would 
take  them  9  times  as  long  to  eat  36  bushels.    Ans.  27  days. 


140  .  KEY.  [part  II. 

16.  If  2  men  spend  12  dollars  in  1  week,  1  man  will 
spend  6  dollars  in  1  week,  and  30  dollars  in  5  weeks,  and 
3  men  would  spend  3  times  as  much,  or  90  dollars. 

17.  The  shadow  of  the  staff  is  f  of  the  length  of  the  staff, 
therefore  the  shadow  of  the  pole  must  be  |  the  length  of  the 
pole.     18  feet  is  |  of  27  feet. 

20.  It  would  take  2  men  3  times  as  long  to  do  it  as  it 
would  6  men. 

23.  8  men  would  do  a  piece  of  work  1  half  as  large  in  2 
days,  and  it  would  take  2  men  4  times  as  long  to  do  it,  or  8 
days. 

28.  He  must  sell  it  for  56  dollars  in  order  to  gain  16  dol- 
lars.    56  dollars  is  7  dollars  per  barrel. 

29.  It  cost  him  36  dollars,  and  he  must  sell  it  for  45  to 
gain  10  dollars ;  45  dollars  is  9  dollars  a  firkin. 

30.  Ans.  56  cents.     See  Section  VI. 

33-  If  it  would  last  3  men  10  months,  it  would  last  1  man 
30  months,  and  5  men  6  months. 

34.  There  are  8  times  5  in  40;  and  since  the  other  would 
build  as  many  times  9  as  the  first  does  5,  he  would  build  6 
times  9,  or  72  rods. 


SECTION  VII. 

A.  13.  i  of  20  is  4,  f  are  16  ;  16  being  ^,  2  is  | ;  2  is  j 
of  14,  and  6  is  f  of  14. 

16.  f  of  28  are  12 ;  12  is  2  times  6,  and  6  is  i  of  48,  (12 
is  I  of  48, )  and  48  is  6  times  seven  and  f  of  7. 

B.  1.  f  of  15  are  12 ;  12  is  6  times  2 ;  2  is  yV  of  20 ;  (12 
is  A  of  20 ;)  i  of  21  is  7  ;  20  is  2  times  7  and  f  of  7. 

2.  I  of  18  are  24 ;  24  is  f  of  27 ;  +  of  35  is  5 ;  27  is  5 
times  5  and  |  of  5. 

C.  This  article  contains  the  multiplication  table,  in  which 
the  numbers  from  10  to  20  are  multiplied  by  the  ten  first 
numbers. 


SECTION  VIII. 

A.  1.  In  one  there  are  2  halves ;  in  2  there  are  twice  as 
many  halves,  that  is  4  halves  or  f . 


SEC.  IX.]  KEr.  141 

2.  In  one  there  are  2  halves,  in  3  there  are  three  times 
two  halves,  that  is,  6  halves,  or  f . 

3.  In  one  there  are  3  thirds,  in  2  there  are  twice  3  thirds, 
that  is,  6  thirds,  or  f. 

15.  Draw  two  lines  on  the  board  thus  ■.    Each  line 

call  one ;  divide  the  upper  line  into  two  equal  parts,  each 
part  is  one  half  of  one,  or  one  half;  divide  the  lower  line 
in  the  same  way,  and  then  call  for  the  answer.  The  form 
of  the  question  may  be  varied  by  asking  how  many  half 
apples  there  are  in  2  apples. 

37.  In  5  and  2  thirds,  how  many  thirds?  Draw  five 
parallel  equal  lines,  divide  them  into  thirds,  and  draw 
another  parallel  with  the  others  and  two  thirds  as  long,  let 
the  scholar  count  the  thirds.  In  all  these  cases  of  illustra- 
tion oh  the  boaixi,  it  is  presumed  that  the  scholar  will  soon 
take  the  hint,  and  devise  illustrations  for  himself  at  his  seat. 


SECTION  IX. 

A.  2.  ^  signifies  that  1  thing  is  divided  into  3  equal 
parts,  and  1  part  taken.  Therefore  2  times  1  third  is  2  parts, 
or  f .      _ 

6.  7  times  ^  is  ^,  or  2}. 

B.  4.  4  times  2  are  8,  and  4  times  1  half  are  4  halves,  or 
2,  which  added  to  8  make  10. 

18.  4  times  3  are  12,  and  4  times  f  are  ^  or  three  whole 
ones,  which  added  to  12  make  15. 

32.  2  timfes  3  are  6,  and  2  times  ^  are  f ,  which  added  to 
6  make  6f . 

40.  10  barrels  at  3  dollars  and  f  a  barrel ;  10  barrels  at 
3  dollars,  would  be  30  dollars,  then  10  times  f  is  *^,  or  8 
and  f  of  a  dollar.     Ans.  38f  dollars. 

C.  2.  I  to  each  would  be  3  times  f ,  or  f ,  which  are  2J 
oranges. 

3.  V*  or  2  bushels. 

4.  7  times  f  are  ^^ ,  or  5^  gallons. 

5.  8  yards  and  f  or  2  yards,  that  is  10  yards. 

6.  4  times  2  are  8,  and  4  times  f  are  ^,  or  2f,  which 
added  to  8  make  lOf  bushels. 


142  KEY.  [part  ii. 

12.  It  would  take  1  man  3  times  as  long  as  it  would  3 
men.     Ans.  13f  da}-,?. 

14.  3  men  would  build  3  times  as  much  as  1  man ;  and 
in  4  days  they  would  build  4  times  as  much  as  in  1  day. 
A)is.  38f  rods. 

15.  A7is»  12  yards. 


SECTION  X. 

A.  21.  |-  of  1  is  i-.  i  of  2  is  2  times  as  much,  or  |.  ^ 
of  4  is  f ,  or  U.  ^  of  5  is  f ,  or  If.  J  of  6  is  f ,  or  2.  l 
of  7  is  I,  or  2i 

27.  J  of  1  is  ^.  ^  of  2  is  f .  ;^  of  3  is  f .  ^  of  7  is  ?, 
or  If. 

This  manner  of  reasoning  may  be  applied  to  any  num- 
ber. To  find  j  of  38:  it  is  3/,  for  j  of  38  is  38  times  as 
much  as  ^  of  1,  and  |  of  1  is  |,  consequently  j  of  38  is  V, 
and  ^'^  is  5f. 

40.  To  find  |-  of  a  number,  -J  must  be  found  first,  and 
then  I  will  be  2  times  as  much.  :J  of  7  is  f ,  and  2  times 
I  are  y,  or  4f. 

74.  ^  of  50  is  ^gO,  or  5f ;  |  is  4  times  as  much;  4  times 
5  are  20,  4  times  f  are  ^JJ,  or  2f,  which  added  to  20  make 
22f. 

Note. — The  manner  employed  in  example  40th  is  best 
f  a-  small  numbers,  and  that  in  the  74th  for  large  numbers. 

B.  2.  Ans.  If  apiece. 

3.  ;J^  of  3  is  f ;  f  of  a  bushel  apiece. 

4.  f  of  7  is  4|;  he  gave  away  4^,  and  kept  2f. 
G.  1  half  dollar  a  yard,  or  50  cents. 

7.  i  of  7  is  I,  or  If ;  |  of  a  dollar  is  f  of  100  cents,- 
which  is  40  cents.     Ans.  1  dollar  and  40  cents  a  bushel. 

8.  i  of_8  is  If;  f  of  100  is  33|.  Ans.  1  dollar  and  33| 
cents,  or  it  is  1  dollar  and  2  shillings. 

9.  If  3  bushels  cost  8  dollars,  1  bushel  will  cost  2  dollars 
and  f,  and  2  bushels  will  cost  5J  dollar's.  Ans.  5  dollars 
and  2  shillings,  or  33|  cents. 

13.  If  7  pounds  cost  40  cents,  1  will  cost  5f  cents;  10 
pounds  will  cost  57^^  cents. 

16,  1  cock  would  empty  it  in  6  hours,  and  7  cocks  would 
empty  it  in  ^  of  6  hours,  or  f  of  1  hour,  which  is  f  of  60 
minutes;  f  of  60  minutes  is  51§  minutes. 


SEC.  XI.]  KEY.  143 

SECTION  XL 

A.  2.  2  halves  of  a  number  make  the  number,  conse- 
quently 1  and  1  half  is  the  half  of  2  times  1  and  1  half, 
which  is  3. 

15,  4f  is  j;  of  5  times  4  and  f ,  which  is  22|. 
17.  4f  is  i  of  9  times  4^,  which  is  39f . 

B.  4,  5  is  3  times  ^  of  5,  which  is  f ,  or  If. 

30.  If  8  is  I  of  some  number,  ^  of  8  is  ^  of  the  same 
number.  ^  of  8  is  2f,  2|  is  1  of  4  times  2f  which  is  lOf; 
therefore  8  is  f  of  lOf. 

40.  If  8  is  f,  h  of  8  is  ,^;  i  of  8  is  |,  f  is  ^  of  V,  or  9| ; 
therefore  8  is  f  of  9|. 

52.  If  f  of  a  ton  cost  23  dollars,  ^  of  a  ton  must  be  |  of 
23,  that  is  4f  dollars,  and  the  whole  would  cost  9  times  as 
much,  that  is  41?, 

09.  ^  of  65  is  7f ;  7|  is  i  of  5  times  7f ,  which  is  36^. 
C5  is  f  of  36^. 

C.  4.  37  is  f  of  32f,  which  taken  from  37  leaves  4|. 
Ans.  4^  dollars. 

5.  7  feet  must  be  |  of  the  whole  pole. 

6.  If  he  lost  f,  he  must  have  sold  it  for  |  of  what  it  cost. 
47  is  I  of  60f .     Ans,  60  dollars  and  42f  cents. 

Miscellaneous  Examples. 

1.  The  shadow  of  the  staff  is  f  of  the  length  of  the  staff; 
therefore  the  shadow  of  the  pole  is  f  of  the  length  of  the 
pole.     67  is  f  of  83 f.   _  Ans.  83 f  feeL 

2.  9  gallons  remain  in  the  cistern  in  1  hour.  It  Avill  be 
filled  in  10  hours  iand  ^ ;  ^  of  60  minutes  are  46|  minutes 
and  I ;  |  of  60  seconds  are  40  seconds.  Aris.  10  hours,  46 
minutes,  40  seconds. 

10.  Find  f  of  33,  and  subtract  it  from  17.     Ans.  3f . 

11.  It  will  take  3  times  10  yards. 

13.  5  is  f  of  3  ;  it  will  take  f  as  much.  Or,  7  yards,  5 
quarters  wide,  are  equal  to  35  yards  1  quarter  wide,  which 
is  equal  to  llf  yards  that  is  3  quarters  wide. 

15.  f  of  37  dollars. 

16.  f  as  much. 


144  KEY.  [part  11. 

SECTION  XII. 

The  examples  in  this  section  are  performed  in  precisely 
the  same  manner  as  those  in  the  sections  to  M^hich  they 
refer.  All  the  difficulty  consists  in  comprehending  that 
fractions  expressed  in  figures  signify  the  same  thing  as 
when  expressed  in  words.  Make  the  pupil  express  them  in 
words,  and  all  the  difficulty  will  vanish.  Let  particular 
attention  be  paid  to  the  explanations  of  fractions  given  in 
this  section. 

VIII.  A.  6.  In  7  how  many  ^?  expressed  in  words,  is 
"in  7  how  many  sixths?     Ans.   */. 

14.  Reduce  8/^  to  an  improper  fraction;  that  is,  in  8 
and  3  tenths  how  many  tenths?     Ans.  f§. 

B.  8.  ^yp  are  how  many  times  1  ?  That  is,  in  23  sev- 
enths how  many  whole  ones  ?     A7is.  3f . 

IX.  B.  3.  How  much  is  5  times  6f  ?  That  is,  how  much 
is  5  times  6  and  4  sevenths?     Ans.  32f. 

V.  &  X.  15.  What  is  |  of  27?  That  is,  what  is  5  eighths 
of  27?     Ans.  16|. 

VI.  &  XI.  A.  8.  7f  is  i  of  what  number?  That  is,  7 
and  6  sevenths  is  1  eighth  of  what  number?     Aois.  62f . 

B.  4.  12  is  f  of  what  number?  That  is,  12  is  3  sevenths 
of  what  number?     Ans.  28-. 

12.  4  is  f  of  what  number?  That  is,  4  is  3  fifths  of 
what  number?     Ans.  6| 


SECTION  XIII. 

The  operations  in  this  section  are  the  reducing  of  frac- 
tions to  a  common  denominator,  and  the  addition  and 
subtraction  of  fractions.  The  examples  will  generally 
show  what  is  to  be  done,  and  how  it  is  to  be  done. 

4.  It  will  readily  be  seen  that  ^  and  ^  are  f . 

25.  In  the  fourth  square  of  the  second  row,  it  will  be 
seen  that  1  half  is  | ;  and  in  the  second  square  of  the 
fourth  row,  j  is  |,  both  together  make  |  and  ^  make  ■^. 

27.  f  is  the  same  as  f . 

When  these  questions  are  performed  in  the  mind,  the 
pupil  will  explain  them  as  follows.  He  will  probably  do 
it  Tvithout  assistance.     Twenty  twentieths  make  one  whole 


SEC.  XIII.]  KEY.  145 

one.  i  of  20  is  5,  and  f  of  20  is  8,  and  ^s  of  20  is  2; 
therefore  J  is  /j,  f  is  ^%,  and  -j^^  is  g^y.  All  the  examples 
should  be  explained  in  the  same  manner, 

45..  One  whole  orte  is  f  g,  one  eighth  of  f  |  is  j^.  |  is  3 
times  as  much,  which  is  |^. 

51.  1  half  is  |,  and  i  is  f,  which  added  together  make  |. 

^1-  f  is  2V.  1^6  i«  2%,  i  is  /ff,  which  added  together 
make  ^^. 

67.  f  is  j8y,  I  is  j%,  which  added  together  make  jj;  from 
fl  take  /j,  and  there  remains  jf,  or  1. 

82.  It  will  be  easily  perceived  that  these  examples  do 
not  dijQTer  from  those  in  the  first  part  of  this  section,  except 
in  the  language  used.  They  must  be  reduced  to  a  common 
denominator,  and  then  they  may  be  added  and  subtracted 
as  easily  as  v.hole  numbers,  f  "is  \^,  and  f  is  -j^j,  and  both 
together  make  ]f,  or  11. 

80.  ^  is  I,  and  '}  is  §.  If  |  be  taken  from  f,  there  re- 
mains i. 

B.  This  article  contains  only  a  practical  application  of 
the  preceding. 

3.  This  example  and  some  of  the  following  contain  mixed 
numbers,  but  they  arc  quite  as  easy  as  the  others.  The 
whole  numbers  may  be  added  separately,  and  the  fractions 
reduced  to  a  common  denominator,  and  then  added  as  in 
other  cases,  and  afterwards  joined  to  the  whole  numbers. 
0  and  2  are  8;  1  half  and  J  are  |,  making  in  the  whole  8f 
bushels. 

^    5.  6  and  2  are  8;  f,  and  ^,  and  f  are  f^,  or  1^|,  which 
joined  with  8  make  9^g. 

C.  It  is  difficult  to  find  examples  which  will  aptly  illus- 
trate this  operation.  It  can  be  done  more  conveniently  by 
tlie  instructor.  Whenever  a  fraction  occurs,  which  may  be 
reduced  to  lower  terms,  if  it  be  suggested  to  the  pupil^  he 
will  readily  perceive  it  and  do  it."  This  may  be  done  in 
almost  any  part  of  the  ■  book,  but  more  especially  after 
studying  the  13th  section.  Perhaps  it  would  be  as  well  to 
omit  this  article  the  first  time  the  pupil  goes  through  the 
book,  and,  after  he  has  $^ea  the  use  of  the  operation,  let 
him  study  it. 


146  KEY.  [part  II. 

SECTION  XIY. 

A.  This  roetion  contams  the  division  of  fractions  "by 
whole  numhei's,  and  the  multiplication  of  one  fraction  hy 
another.  Though  these  operations  sometimes  appear  to  be 
division,  and  sometimes  multiplication,  yet  there  is  actually 
no  difference  in  the  operations. 

The  practical  examples  will  generally  show  how  the 
operations  are  to  he  performed,  hut  it  will  be  well  to  illus- 
trate the  operation  for  young  pupils. 

1  and  2.  ^  of  ^^  is  i  of  the  whole. 

3  and  4.  |  of  ^  is  ^. 

16  and  17.  ^  of  |  is  y^  of  the  whole. 

33.  Since  |  of  a  share  signify  3  parts  of  a  share,  it  is 
evident  that  ^  of  the  three  parts  is  1  part,  that  is,  |-. 

39.  f  signify  9  pieces  or  parts,  and  it  is  evident  that  ^ 
of  9  parts  is  3  parts,  that  is  f . 

43.  We  cannot  take  |  of  5  pieces,  therefore  we  must  take 
i  of  i,  which  is  j'g ,  and  f  is  5  times  as  much  as  i,  therefore 
i  of  f  is  j-V- 

78.  8f  is  V-iof  V'isl. 

79.  8f  is  *V',  I  of  I  is  J^,  consequently,  ^  of  ^  is  f  §,  or 
111. 

*86.  We  may  say  ^  of  8|  is  2,  and  2|  over,  then  2|  is  %^, 
and  i  of  y  is  f f ,  hence  ^  of  8|  is  2|f . 

90.  ^  of  18f  is"2ff,  and  f  is  3  times  as  much,  or  7ff. 

B.  4.  It  would  take  1  man  4  times  9f ,  or  37f  days,  and 
7  men  would  do  it  in  |  of  that  time,  that  is,  in  5^f  days. 


SECTION  XV. 

A.  This  section  contains  the  divisions  of  whole  numbers 
by  fractions,  and  fractions  by  fractions.  - 

1.  Since  there  are  f  in  2  it  is  evident  that  he  could  give 
them  to  G  boys  if  he  gave  them  J  apiece  ;  but,  if  he  gave 
them  f  apiece,  he  could  give  them  to  only  one  half  as  many, 
or  3  boys.  » 

5,  If  -J^  of  a  barrel  would  last  them  one  month,  it  is  evi- 
dent that  4  barrels  would  last  20  months  ;  but  since  it  takes 
I  of  a  barrel,  it  will  last  them  but  one  half  as  long,  or  10 
months. 


SEC.  XV.]  KEY.  147 

7.  6|  is  y.  If  I  of  a  bushel  would  last  a  week,  6| 
bushels  would  last  27  Aveeks  ;  but,  since  it  takes  f  it  will  last 
only  J  of  the  time,  or  9  weeks. 

13.  If  ho  had  given  j  of  a  bushel  apiece,  he  might  have 
given  it  to  17  persons ;  but,  since  he  gave  3  halves  apiece, 
he  could  give  it  to  only  ^  of  that  number,  that  is,  to  5  per- 
sons, and  he  would  have  1  bushel  left,  which  would  be  f  of 
enou2:h  for  another. 

23!  Of  is  V'  {^n^  U  is  V-  If  it  had  been  onljit  of  a 
dolhir  a  barrel,  he  might  have  bought  66  barrels  for  9f  dol- 
lars ;  but,  since  it  was  V  a  barrel,  he  could  buy  only  ^j  of 
that  numlier,  that  is,  6  barrels.    • 

25  and  26.  Ans.    9  4. 

31.  4|  is  ^^ ,  and  9f  is  ^.  Now  ^  is  contained  in  Y 
48  times,- and  V  is*  contained  only  jy  P^^t  as  many  times, 
consequently  only  25^  or  2f . 

B.  1.  J  is  /jj ;  consequently  5  pounds  can  be  bought  for 
J^  of  a  dollar. 

3.  f  is  -fr^,  and  \  is  j\.  If  he  had  given  only  -^  apiece, 
he  could  have  given  it  to  9  persons;  but  since  he  gave  -^^ 
he  could  give  it  to  only  1  half  as  many,  or  4^  persons. 

5.  if  is  j^\,  and  f  is  |f .  If  a  pound  had  cost  j'y  of  a  dol- 
lar, 14  pounds  could  be  bought  for  |f  of  a  dollar ;  but 
since  it  costs  /y,  only  |  as  many  can  be  bought ;  that  is,  4f 
pounds. 

9.  f  is  \l,  and  If  is  f§.  If  a  bushel  had  cost  5^  of  a 
dollar,  65  bushels  might  have  been  bought ;  but,  since  it 
cost  ^^,  only  ^'g^  part  as  much  could  be  bought;  that  is  4^*5- 
bushels. 

12.  I  is  j^g,  and  f  is  jf  ;  ^s  i^  contained  in  |f  15  times, 
but  -^  is  contained  only  \  as  many  times  ;  that  is,  3|  times. 


MISCELLANEOUS   EXAMPLES. 

5.  f  of  a  penny  is  f  of  4  ftirthings.     Ans.  2f  farthings. 

6.  f  of  12  pence.     Ans.  10  pence. 

7.  f  of  4  quarters  is  2  quarters  and  f  of  a  quarter ;  f 
of  a  quarter  is  |  of  4  nails,  which  is  If  nails.  Ans.  2 
quarters.  If  nails. 

13.  f  of  24  hours  is  15  hours. 

14.  I  of  24  hours  is  14  hours  and  f  of  an  hour ;  |  of  60 
minutes  is  24  minutes.    Ans.  14  hours,  24  minutes. 


148  KEY.  PART  II, 

28.  There  "beinp;  4  farthings  in  a  penny,  1  farthing  is  ^ 
part  of  a  penny. 

30.  3  farthings  is  f  of  a  penny. 

31.  1  penny  is  -^^  of  a  shilling,  because  there  are  12 
pence  in  a  shilling. 

34.  5  pence  is  /^  of  a  shilling. 
41.  1  shilling  is  ^V  ^^  ^  pound. 
43.  3  shillings  is  /^  of  a  pound. 

48.  1  farthing  is  ^^  of  one  shilling. 

49.  2  farthings  is  ^^g,  or  ^^  of  a  shilling.  5  farthings  is 
/^  of  a  shilling. 

51.  1  penny  is  ^Ij^  of  1  pound.     7  pence  is  j^^  of  i£l. 
59.  3s.  5d.  is  41  pence,  which  is  ^^^  of  £1. 
75.  1  nail  is  j'g^  of  a  yard.     5  nails  is  /^  of  a  yard. 
89.  1  oz.  is  tV  of  1  lb.     15  oz.  is  |f  of  1  lb. 
91.  1  lb.  is  i^-g  of  1  quarter.     9  lbs.  is  /^  of  1  quarter. 
100.  At  the  end  of  1  hour  they  would  be  7  and  f  miles 
■  apart.     In  7  hours,  7  times  7f ,  Avhich  is  54f  miles. 

121.  This  is  the  principle  of  fellowship ;  3  shillings  were 
paid  ;  one  paid  i,  the  other  f . 

122.  One  paid  |,  the  other  |.  ""    . 

123.  20  dollars  were  paid  in  the  whole;  one  paid  ^^, 
another  /j^,  and  the  third  /(^. 

126.  3,  and  4,  and  5  are  12.  The  first  put  in  -^^ ;  the 
second  j\ ;  the  third  /g-* 

129.  4  dollars  for  2  months  is  the  same  as  8  dollars  for  1 
month ;  3  dollars  for  3  months  is  the  same  as  9  dollars  for 
1  month ;  and  2  dollars  for  4  months  is  the  same  as  8  dol- 
lars for  1  month.  The  question  is  the  same  as  if  A  had 
put  in  8  dollars,  B  9  dollars,  and  C  8  dollars.  A  must 
have  /j,  B  j^-g,  and  C  j\  of  100  dollars. 

131.  A's  money  was  in  4  times  as  long  as  C's.  It  is  the 
same  as  if  A  had  put  in  8  dollars  for  the  same  time,  and  B 
8  dollars  for  the  same  time.  A  must  have  /j,  B  -2^,  and  C 
2^  of  88  dollars. 

The  examples  127,  128,  129,  130,  and  131,  are  double  or 
compound  foUowship. 

139.  The  interest  of  50  dollars  for  1  year  and  6  months 
is  4  dollars  and  50  cents,  and  for  1  month  it  is  25  cents. 
The  interest  of  7  dollars  for  19  months  (a  dollar  is  J  of  a 
cent  a  month)  is  66j  cents.  The  whole  amounts  to, 5  dol- 
lars and  41 1  cents. 

140.  The  interest  of  200  dollars  for  1^  year  is  16  dol- 
lars. The  interest  of  67  dollars  is  67  cents  for  every  2 
months,  for  16  months  it  will  be  8  times  67  cents,  which 


SEC.  XV.]  KEY.  149 

are  5  dollars  and  36  cents.     The  whole  interest  is  21  dol- 
lars and  36  cents, 

143.  The  interest  of  100  dollars  for  2}  years  is  13  dollars 
and  50  cents.  The  interest  of  100  dollars  for  60  days 
would  be  1  dollar,  the  interest  for  20  days  will  be  i  of  a 
dollar,  or  33^  cents.  The  interest  of  1  dollar  for  2;^  years 
is  13k  cents ;  for  10  dollars  the  interest  would  be  1  dollar 
and  35  centa,  and  for  30  dollars,  4  dollars  and  5  cents. 
The  interest  of  7  dollars  for  21  years  is  7  times  13^  cents, 
or  94J  cents.  The  interest  of  3Y  dollars  for  60  days  would 
be  37  cents,  and  for  20  days  i  of  37  cents,  or  12i  cents. 
The  whole  interest  is  18  dollars  and  95^  cents. 

146.  They  would  both  together  do  |  of  the  work  in  1 
day,  and  it  would  take  them  ^  of  a  day  to  do  the  other  i. 
Ans.  1^  day. 

149.  They  both  together  consume  J  of  a  bushel  in  a 
week,  but  the  woman  alone  consumes  only  ^  of  a  bushel  in 
a  week.  That  is,  they  both  together  consume  /j  in  a  week, 
but  the  woman  alone  only  j*jj ;  consequently  the  man  alone 
would  consume  ^^^ ;  and  a  bushel  would  last  him  3  J  weeks. 

152.  A  and  B  can  build  i  of  it  in  1  day ;  A,  B,  and  C, 
can  build  |  of  it  in  1  day ;  4he  difference  between  ^  and  ^ 
is  ^Q ;  therefore  C  can  build  -^j^  of  it  in  1  day ;  and  it  would 
take  him  13i  days  to  build  it  alone. 

164.  Find  how  much  they  might  eat  in  a  day,  in  order  to 
make  it  last  1  month,  and  then  it  will  be  easy  to  find  how 
much  they  may  eat  in  a  day,  to  make  it  last  11  months. 

167.  The  money  is  7  parts  of  the  whole,  and  the  purse 
1  part ;  consequently  the  money  is  |  and  the  purse  i  of  16. 

170.  He  gave  1  part  for  the  apple,  2  parts  for  the  orange, 
and  4  parts  for  the  melon.  These  make  7  parts.  The 
apple  3  cents,  the  orange  6  cents,  and  the  melon  12  cents. 

175.  If  to  a  number  half  of  itself  be  added,  the  sum  is 
I  of  that  number ;  hence  subtract  2J  from  100,  and  the 
remainder  is  f  of  the  number  of  geese  that  he  had. 

180.  This  must  be  reduced  to  6ths.  1  half  is  f ,  and  i 
is  f ,  and  the  number  itself  is  f .  If  therefore  to  the  whole 
number  its  half  and  its  third  be  added,  the  sum  will  be 
y  ;  hence,  77  is  y  of  the  number. 

181.  i  is  f ;  therefore  if  to  a  number  h  and  i  of  itself 
be  added,  the  whole  number  will  be  I ;  but  when  18  more 
is  added  to  ^,  the  first  number  is  doubled;  that  is,  the 
number  is  f  of  the  first  number ;  therefore  18  is  J  of  the 
number. 


AEITHMETIC. 


PART    III. 


WRITTEN    ARITHMETIC 


NOTATION    AND    NUMERATION. 

1.  Instead  of  writing  the  names  of  numbers,  it  is 
■usual  to  express  them  by  particular  characters,  called 
figures. 


One                 is 

written 

1 

Twenty-six  is  written 

.     26 

Two 

li 

2 

Twenty-seven 

a 

27 

Three 

a 

3 

Twenty-eight 

li 

28 

Four 

(( 

4 

Twenty-nine 

11 

29 

Five 

(( 

5 

Thirty 

.11 

30 

Six 

a 

6 

Thirty-one 

11 

31 

Seven 

(( 

7 

Forty 

11 

40 

Eight 

it 

8 

Fifty 

(( 

50 

Nine 

a 

9 

Sixty 

i< 

60 

Ten 

a 

10 

Seventy 

i( 

70 

Eleven 

(( 

11 

Eighty 

li 

80 

Twelve 

i( 

12 

Ninety 

(I 

90 

Thirteen 

a 

13 

One  hundred 

li 

100 

Fourteen 

i( 

14 

One  hundred  and 

Fifteen 

i( 

15 

one 

11 

101 

Sixteen 

li 

16 

One  hundred  and 

Seventeen 

li 

17 

two 

li 

102 

Eighteen 

a 

18 

One  hundred  and 

Nineteeh 

a 

19 

ten 

u 

110 

Twenty 

a 

20 

One  hundred  and 

Twenty-one 

11 

21 

twenty 

11 

120 

Twenty-two 

a 

22 

Two  hundred 

a 

200 

Twenty-three 

a 

23 

Three  hundred 

11 

300 

Twenty-four 

a 

24 

One  thousand 

11 

1000 

Twenty-five 

iC 

25 

Ten  thousand 

'I  10,000 

PART  III.]  ARITHMETIC.  151 

2,  When  several  figures  stand  side  by  side,  the 
value  of  each  figure  depends  on  its  place  as  counted 
from  the  right  towards  the  left.  A  figure  standing  in 
the  first  place,  signifies  so  many  units,  or  ones;  the 
same  figure,  in  the  second  phice,  signifies  so  many  fens; 
in  the  third  place,  hundreds;  in  the  fourth  place, 
thousands ;  in  the  fifth  place,  fens  of  thousands ;  and 
in  the  sixth  place,  hundreds  of  thousands.  Therefore 
every  time  a  figure  is  removed  one  place  to  the  left,  its 
value  is  made  tenfold,  as  may  be  seen  in  the  following 
table : 


03 

o-^ 

'^  a  '^    ' 

"S  a  o  «  c-"^ 

W^hwhp  [hundred  and  eleven. 

111^111,  read  One  hundred  and  eleven  thousand,  one 
4    "    Four. 
49     "    Forty-nine. 
496    "    Four  hundred  and  ninety-six.  [five. 

4,965    "    Four  thousand  nine  hundred  and  sixty- 
49,652    ^'    Forty-nine  thousand   six    hundred   and 
fifty-two.        ^    • 
496,527    '^    Four  hundred  and  ninety-six  thousand 
five  hundred  and  twenty-seven. 
S.     Let  the  pupil  read  the  following  numbers: 


1. 

34 

5. 

142 

9. 

6,744 

13. 

11,504 

2. 

76 

6. 

279 

10. 

8,329 

14. 

50,603 

3. 

68 

7. 

758 

11. 

4,960 

15. 

124,769 

4. 

95 

8. 

423 

12. 

9,080 

16. 

784,984 

4,     Let  the  pupil  express  the  following  numbers  in 
figures  : 

1.  Seventy-eight. 

2.  Ninety. 


152.  ARITHMETIC.  [part  iii. 

3.  One  hundred  and  fifty-six. 

4.  Three  hundred  and  eight. 

5.  One  thousand  four  hundred  and  sixty-nine. 

6.  Five  thousand  and  two. 

7.  Seventy-seven  thousand  and  eighty-four, 

8.  Forty  thousand  seven  hundred  and  twenty-two. 

9.  Five  hundred  and  sixty-six  thousand  one  hundred 

and  fifty. 
10.  Six  hundred  thousand. 

ADDITION. 

5.  5  and  4  and  7  are  how  many  ?  This  is  called 
a  question  in  addition.  We  put  together,  or  add  the 
numbers  5,  4  and  seven,  to  find  their  sum  or  amount. 
Thus,  the  sum  of  5  and  4  and  7  is  16. 

Ex.  1.  What  is  the  sum  of  46  and  37? 
gi  It  is  the  most  convenient  to  write  the  numbers 

^;§  in  order,  units  under  units,  tens  under  tens,  etc.; 
46  then  besinnino;  with  the  units,  we  find  that  6 
37  units  and  7  units  are  13  units,  or  1  ten  and  3 
—  units.  We  write  down  the  3  units  under  the 
83  units  and  add  the  1  ten  to  the  other  tens;  1  (ten) 
and  4  (tens)  and  3  (tens)  make  8  (tens)  which 
written  with  the  3,  makes  83. 

Ex.  What  is  the  sum  of  736  and  875  and  394? 
.§  Adding  the' units  we  find  6  and  5  are  11,  and 

^  .  ^  4  are  15,  or  1  ten  and  5  units.  Writing  the 
s  g  S  5  under  the  column  of  units,  wc  reserve  the  1 
^^P  ten,  and  add  it  to  the  tens  of  the  next  column; 
i.-r  thus,  1  and  3  are  4,  and  7  are  11,  and  9  are  20 
7,  La      (tens)  or  2  hundreds,  and  0  tens.'    We  write 

2. the  0  under  the  column  of  tens,  and  add  the 

200  5  2  (hundreds)  to  the  column  of  hundreds;  thus, 
2  and  7  are  9,  and  8  are  17,  and  3  are  20  (hundreds,) 
which  being  the  sura  of  the  last  column  we  write  down. 
The  answer,  therefore,  is  2005. 

6.  Add  the  following : 


PAKT  III.] 

ARITHMETIC. 

163 

3. 

4 

5. 

6.      7. 

8. 

9. 

10. 

11. 

15 

33 

8 

7    19 

28 

45 

56 

69 

4 

5 

9 

5    5 

7 

8 

7 

5 

12. 

lo. 

14. 

15.     16. 

17. 

18. 

19. 

20. 

85 

17 

75 

35   62 

84 

93 

60 

80 

42 

50 

26   79 

45 

58 

30 

50 

21. 

22. 

23. 

24.     : 

i5. 

26. 

27. 

28. 

27 

60 

47 

234   311 

435 

731 

478 

32 

88 

55 

362   5 

25 

924 

311 

298 

56 

99 

72 

403   293 

808 

579 

527 

29. 

30. 

31. 

32. 

3 

3. 

34. 

500 

7667 

7543 

1812 

4648 

789 

314 

1: 

235 

1858 

1732 

4e 

120 

2253 

409 

5555 

1776 

2712 

8665 

226 

1492 

37. 

3333 

452 

45 

36. 

35. 

38. 

39. 

990 

8 

6540 

38,924 

468 

6209 

295 

6030 

67,798 

26,792 

2316 

17 

2709 

54,554 

15,864 

2201 

40. 

7897 

5967 
together  39, 

67,1; 

33,333 

^4,  51 

80,112 

7. 

Add 

12,  43,  I 

41. 

Add  together  27C 

),  489 

,  126,  46.' 

42. 

Add  together  2368,  23 

5,  496, 

1736. 

43. 

Add  together  66,476,  8 

01,400! 

9,32, 

127. 

8.  44.  A  merchant  bought  4  pieces  of  cloth  ;  for 
the  first  piece  he  gave  53  dollars;  for  the  second,  78 
dollars;  for  the  third,  68  dollars;  and  for  the  fourth, 
67  dollars.     How  much  did  he  give  for  the  whole  ? 

45.  In  an  orchard  there  are  22  peach  trees,  95  apple 
trees,  17  pear  trees,  and  56  plum  trees.  How  many 
trees  in  the  orchard? 

46.  A  man  bought  four  horses;  for  the  first  he  gave 
84  dollars;  for  the  second,  150  dollars;  for  the  third, 


154  ARITHMETIC.  [part  iil 

475   dollars;   and  for  the  fourth,  526   dollars.     How 
much  did  he  give  for  all  the  horses  ? 

47.  In  Mr.  Green's  school  there  are  810  scholars  ; 
in  Mr.  White's,  697  scholars;  in  Mr.  Brown's,  956 
scholars;  in  Mr.  Blue's,  776  scholars;  in  Mr.  Black's, 
572  scholars;  and  in  Mr.  Crimson's,  731  scholars.  How 
many  scholars  in  these  six  schools  ? 

48.  A  gentleman,  building  a  house,  paid  the  mason 
2,964  dollars,  the  carpenter  5,723  dollars,  the  plasterer 
625  dollars,  and  the  painter  354  dollars.  How  much 
did  the  whole  house  cost  ? 

SUBTRACTION. 

O.  9  less  seven  are  how  many  ?  This  is  called  a 
question  in  Subtraction.  We  take  away,  or  subtract, 
7  from  9  to  find  the  difference  or  remainder,  which  is  2. 

Ex.  1.  From  476  take  245. 
^  It  is  most  convenient  to  write  the  less  num- 

^  .  ^  ber  under  the  greater,  units  under  units,  tens 
s£a  under  tens,  etc.,  as  in  addition.  Beginning  at  the 
'^^^  units,  we  say  5  from  6  leaves  1,  which  we  write 
^  ^  "  under  the  units ;  then  4  (tens)  from  7  (tens) 
*^^^  leaves  3  (tens,)  which  we  write  under  the  tens; 
and  2  (hundreds)  from  4  (hundreds)  leaves  2 
^  ^  -^  (hundreds,)  which  we  write  down,  and  the 
answer  is  231. 

Ex.  2.  From  939  take  458. 
4  In  this  example  we  take  8  from  9  and  there 

^  .  .  remains  1,  which  we  write  down  ;  but  since  we 
cgS  cannot  take  5  (tens)  from  3  (tens),  we  borrow 
^^^  1  (hundred,)  or  10  (tens)  from  the  9  (hundreds,) 
9  3  9  which  joined  with  the  3  (tens,)  makes  13  (tens ;) 
45  8  then  5  (tens)  from  13  (tens)  leaves  S  (tens,) 
— —  which  we  write  down.  As  one  of  the  9  (hun- 
481  drcds)  has  been  put  with  the  3  (tens,)  there 
^  remains  but  8  (hundreds.)     Therefore  we  say  4 

(hundreds)   from  8   (hundreds)  leaves  4  (hundreds.) 
This  written  down  makes,  the  answer  481. 


PART  II] 

•] 

AIIITIIMETIC. 

155 

3. 

4. 

5. 

6. 

fr 

8. 

9. 

From 

18 

58 

35 

59 

68 

82 

95 

Take 

4 

6 

12 

33 

34 

51 

75 

Ans. 

ll 

26 

20 

10. 

11. 

12. 

13. 

14. 

15. 

From 

285 

467 

9685 

7856 

8744 

2607 

Take 

74 

16. 

46 

17. 

5473 

5722 

20. 

4302 

1405 

18. 

19. 

21. 

22. 

From 

12 

13 

19 

16 

24 

56 

92 

Take 

G 

5 

9 

7 

,8 

7 

4 

From 
Take 


30. 

134 
25 


31. 

244 
73 


32 

252 
171 


33. 

888 
459 


34. 

2236 
1145 


35. 

3456 
2246 


23. 

24. 

25. 

26. 

27. 

28. 

29. 

From 

30 

53 

46 

33 

64 

92 

84 

Take 

20 

10 

27 

25 

29 

88 

59 

36. 

9927 
6090 


37.  From  96  take  58. 

38.  From  4785  take  3679. 

39.  From  8344  take  5324. 

40.  From  10982  take  6470^ 


41.  From  984  take  176. 

42.  From  4788  take  967. 

43.  From  7898  take  6898. 

44.  From  40816  take  5622. 


10.  45.  In  a  certain  garden  there  were  86  rose 
bushes,  but  the  worms  destroyed  33  of  them.  How 
many  were  left  ? 

46.  A  man  is  66  years  of  age,  and  his  son  22.  How 
old  was  the  man  at  the  birth  of  his  son  ? 

47.  A  farmer  had  240  sheep  -,  but,  one  night,  wolves 
gained  admittance  to  the  fold,  and  carried  off  121.  How 
many  sheep  were  left? 

48.  Harry  had  84  cents  given  him  to  buy  some  mar- 
bles with,  but  on  his  way  to  the  store  he  lost  25  cents. 
How  many  cents  had  he  left  ? 

49.  A  girl  was  carrying  72  eggs  to  market,  but  she 
let  the  basket  fall,  and  39  eggs  were  broken.  How 
many  had  she  left  ? 


156  ARITHMETIC.  [part  hi. 

50.  In  an  army  there  were  14,642  men,  but  in  a 
battle  789  men  were  killed  or  wounded.  How  many 
serviceable  men  were  there  left  ? 

MULTIPLICATION. 

11.  7  times  5  are  how  mamy  ?  This  is  called  a 
question  in  multiplication.  To  obtain  the  answer  we 
repeat  or  multiply  the  number  5^  7  times;  thus,  7  times 
5  are  35. 

Ex.  1.  Multiply  5G4  by  7. 
^  Writing  the  numbers  thus,  the  7  under  the 

I  .  ^.  units,  we  say,  7  times  4  (units)  are  28  (units,) 
§  1 1  or  2  tens  and  8  units.  We  write  down  the  8 
^^^  (units)  and  reserve  the  2  (tens;)  then  7  times 
5  6  4  6  (tens)  are  42  (tens)  and  the  2  (tens)  which 
7  we  reserved  are  44  (tens)  or  4  hundreds  and 
4  tens.    We  write  the  4  (tens)  and  reserve  the 


3  948     4  (hundreds;)  then,  7  times  5  (hundreds)  are 

35  (hundreds,)  which,  with  the  4  (hundreds) 

we  reserved,  are  39  (hundreds)  or  3  thousands  and  9 

hundreds.    This  we  write  down,  and  the  answer  is  3948. 


Multiply 

By 

8         9 
3         5 

13    ,  : 

2 

23 
3 

43         51 
6          9 

65 

7 

Multiply 
By 

9.                10. 

143      424 

2          4 

11. 
511 

7 

12.                13. 

821        410 
6            5 

14. 

821 
4 

Multiply 
By 

15. 

813 
3 

10. 

962 

8 

17. 

1243 
2 

18. 

2104 
4 

Multiply 

By 

19. 

5134 
6 

•  20. 
7110 

7 

21. 
8947 
9 

22. 
4355 
8 

PART  III.]  ARITHMETIC.  157 

23.  Multiply  129  by  35. 

4  ^  Writing  the  numbers  in  order,  units  under 

I  £     .  units,  and  tens  under  tens,  we  multiply,  first, 

ocS|  by  the  5   units,  and  write  the  result  in   its 

hWh;:^  proper  place,  as  when  we   multiplied  by  one 

;^  2  9  figure.     We  multiply  next  by  the  3   (tens) 

3  5  and  write  the  first  figure  of  the  result  in  the 

. place  of   tens,  because   units  multiplied   by 

(3  4  5  tens  ought  to  produce  tens.     Then  adding 

3  8  7  together  the  results  obtained  by  multiplying 
by  the  5  (units  )  and  3  (tens,)  their  sum  will 

4  515  be  4515,  which  is  the  answer, 

24.  25.  26.  27,  28.  29.  30. 

Multiply       34       35       48       45       99       012       844 
By  23       25       32       44       56         54         29 

102 

68 

Ans.  782 

31,  32.  33.  34.  35.  36.  37. 

Multiply    676     434     467     548     1835     2972     5963 
By  76     322     212     403       768       256     4678 


38.  Multiply  816  by   6,  142,  Multiply  2523  by    47. 

39.  Multiply  363  by  48.  43.  Multiply    682  by  746. 

40.  Multiply  846  by  65,  44.  Multiply  8422  by  186. 

41.  Multiply  152  by  87.  I  45.  Multiply  3107  by  761. 

46.  How  much  will  48  barrels  of  flour  cost  at  8  dol- 
lars a  barrel  ? 

47.  In  a  room  there  are  9  windows  with  16  panes 
of  glass  in  each.  How  many  panes  of  glass  in  all  the 
windows? 

48,  If  a  man  receives  47  dollars  a  month,  how  much 
will  he  receive  in  twelve  months  ? 

49,  In  a  certain  regiment  there  are  13  companies, 
and  126  men  in  each  company.  How  many  men  in 
the  regiment  ? 


153  ARITHMETIC.  [paut  hi. 

50.  There  are  365  days  in  a  j'car.  How  many  days 
in  19  years  ? 

DIVISION. 

IS.  24  are  liow  many  times  8?  This  is  •cninu  a 
question  in  division  ,  because  we  divide,  or  separate, 
the  larger  number  into  as  many  equal  parts  as  there  are 
uuits  in  the  smaller  number. 

13.  The  number  tp  be  divided  is  called  the  divi- 
dend ;  the  number  by  which  we  divide,  the  divisor  ; 
and  the  result,  the  quotient.  If  anything  is  left  after 
the  division,  it  is  called  the  remainder. 

14.  Ex.  1.  Divide  584  by  4. 

Divisor,  4)  584  Dividend.       We  Write  down  the  dividend, 

584,  and  draw  a  line  beneath 

146  Quotient.  it.  Write  the  divisor,  4,  at  the 
left  of  the  dividend,  and  draw  a 
curved  line  between  them.  We  first  find  how  many  times 
4  is  contained  in  5,  the  first  figure  of  the  dividend  ;  it 
is  contained  1  time  and  there  is  1  remainder.  We  write 
the  1  under  the  5,  and  suppose  the  remainder,  1,  written 
at  the  left  of  the  next  figure,  8,  making  the  number  18. 
4  is  contained  in  18,  4  times,  with  2  remainder.  The 
4  we  write  below,  and  suppose  the  2  written  before  the 
next  figure,  4,  making  it  24.  4  is  contained  in  24,  6 
times,  without  remainder.  We  write  the  6  underneath, 
and  the  division  is  completed. 

Division  performed  in  this  way  is  called  slioi-t  divi- 
sion. 

15.  Perform  in  this  way  the  following  examples  : 

2.  3.  4.  5.      '      6.  7. 

Divisor,  3)9  Dividend.       2)26    3)36   4)88     5(15  3)18 

3  Quotient. 
8.  9.  10.  11.  12.  13. 

7(21         9)36         6)48         8)72         9)81         4)40 


PAKT  III.]  ARITHMETIC.  159 

14.  15.  16.  17.  18.  19. 

6)72       2)444       4)256       3)963       7)847       9)279 

20.  21.  22.  23.  24. 

5)6825        6)4650        9)9972        4)4484        8)1312 

10.  When  the  divisor  consists  of  two  or  more 
figures,  it  is  more  convenient  to  perform  the  operation 
in  the  following  manner  : 

25.  Divide  6432  by  12. 
Divisor.  Dividend.  Quotient.      Write  the  dividend,  and  draw 
12)6432(536    a  curved  line  on  both  sides.  We 
6  0  first  inquire  how  many  times  12 

is  contained  in  64,  the  first  two 

4  3  figures  of  the  dividend.  We  find 

3  6  that  it  is  contained  5  times.   We 

place  the   5  at  the  right  of  the 

72  dividend,  as  the  first  figure  of 

7  2  the  quotient,  and  then,  multiply- 

—  ing  the   divisor,    12,   by  5,  we 

obtain  the  number  60,  which  we  write  under  the  64  of 
the  dividend,  and  subtract.  To  the  remainder,  4,  we 
annex  3,  the  next  figure  of  the  dividend,  and  find  how 
many  times  the  divisor,  12,  is  contained  in  43.  We 
place  the  result,  3,  at  the  right  of  the  5  in  the  quotient, 
and,  multiplying  the  divisor  by  3,  we  obtain  36,  which 
we  write  under  the  43,  and  subtract.  To  the  remainder, 
7,  we  annex  the  2  of  the  dividend,  and  find  how  many 
times  the  divisor  is  contained  in  72,  which  is  6  times. 
We  write  6  in  thQ  quotient,  and,  as  there  is  no  re- 
mainder, the  division  is  complete. 

Division  performed  in  this  way  is  callled  long  division. 
17.     Perform  in  this  way  the  following  examples : 

26.  27.  28.  29. 

13)286(22  15)235(  17)425(  13)585( 

26 

"26 
26 


160 

ARITHI\IETIC. 

[part  III. 

30. 

16)X168( 

31.                                  32, 

13)676(          12)1728( 

S3. 

15)2445( 

13)36l6( 

35.                                   36. 

17)2057(          19)2994( 

37. 

15;4445( 

38.  Divide  144  by  12. 

39.  Divide  224  hv  r>. 

40.  d; 

bouclit 

45.  l.": 
acres  can  yo 

46.  If  a  man  c. 
will  it  take  liim  to  ;v 

47.  A  man    divided 


41.  Divide  5456  by  22. 

42.  Divide  1411  by  36. 

43.  Divide  8844  by  22. 

'jts  a  pound  can  be 
iars,  how  many 
day,  how  long 


■"6  m;ies? 
23,796   dollars   among    his   6 
children.     How  much  did  each  receive? 

48.  A  railroad  company  employed  25  workmen,  and 
at  the  end  of  two  months  it  took  1825  dollars  to  pay 
them.     How  much  did  each  man  receive  ? 


THE    END. 


JOSEPH  ftUZtoM 

800KBl«Ct«3 

I  Salt  I  MORE   a.  o  ' 


